21:I[166699,["/_next/static/chunks/2lo1egizl1har.js","/_next/static/chunks/20-st4r-vmt_9.js","/_next/static/chunks/17wissy38xpia.js","/_next/static/chunks/2lra5r_l71ju3.js","/_next/static/chunks/15o6_p55g4jlt.js","/_next/static/chunks/0xr0_zqqxl0i7.js","/_next/static/chunks/0zvch8mjkmtgu.js","/_next/static/chunks/2xi_wpk87ncci.js","/_next/static/chunks/290ytcwgjf1v4.js","/_next/static/chunks/09jx489or_t-1.js","/_next/static/chunks/0-qqdfbk04_51.js","/_next/static/chunks/38j1pvyhos24t.js","/_next/static/chunks/3s82z7gnp_gmt.js","/_next/static/chunks/0700tvgdr9fs0.js"],"NextIntlClientProvider"] 23:I[653641,["/_next/static/chunks/2ios_6dp2jutl.js"],"IconMark"] 24:I[822135,["/_next/static/chunks/2lo1egizl1har.js","/_next/static/chunks/20-st4r-vmt_9.js","/_next/static/chunks/17wissy38xpia.js","/_next/static/chunks/2lra5r_l71ju3.js","/_next/static/chunks/15o6_p55g4jlt.js","/_next/static/chunks/0xr0_zqqxl0i7.js","/_next/static/chunks/0zvch8mjkmtgu.js","/_next/static/chunks/2xi_wpk87ncci.js","/_next/static/chunks/290ytcwgjf1v4.js","/_next/static/chunks/09jx489or_t-1.js","/_next/static/chunks/0-qqdfbk04_51.js","/_next/static/chunks/38j1pvyhos24t.js","/_next/static/chunks/3s82z7gnp_gmt.js","/_next/static/chunks/0700tvgdr9fs0.js","/_next/static/chunks/2pzt8__sf3khr.js","/_next/static/chunks/2qmp8ez7f5-ts.js","/_next/static/chunks/3ac71bzdb8krl.js","/_next/static/chunks/2st4jg72-lc-t.js","/_next/static/chunks/3qss25dcigd49.js","/_next/static/chunks/0wod3pr029w__.js","/_next/static/chunks/0luq3wvp5mo1j.js","/_next/static/chunks/3nal99ptqiuu0.js","/_next/static/chunks/0eh37xdptdijx.js","/_next/static/chunks/03zvd0f1-phhi.js","/_next/static/chunks/1raen7xpdk-3t.js","/_next/static/chunks/1qgibnpd1s4yj.js","/_next/static/chunks/3-3x_d1jt_vz5.js","/_next/static/chunks/3pxd56pjid-ea.js","/_next/static/chunks/2ofzleruxyw3g.js","/_next/static/chunks/1wrt-ta9s4-fa.js"],"Link"] 25:I[323389,["/_next/static/chunks/2lo1egizl1har.js","/_next/static/chunks/20-st4r-vmt_9.js","/_next/static/chunks/17wissy38xpia.js","/_next/static/chunks/2lra5r_l71ju3.js","/_next/static/chunks/15o6_p55g4jlt.js","/_next/static/chunks/0xr0_zqqxl0i7.js","/_next/static/chunks/0zvch8mjkmtgu.js","/_next/static/chunks/2xi_wpk87ncci.js","/_next/static/chunks/290ytcwgjf1v4.js","/_next/static/chunks/09jx489or_t-1.js","/_next/static/chunks/0-qqdfbk04_51.js","/_next/static/chunks/38j1pvyhos24t.js","/_next/static/chunks/3s82z7gnp_gmt.js","/_next/static/chunks/0700tvgdr9fs0.js","/_next/static/chunks/2pzt8__sf3khr.js","/_next/static/chunks/2qmp8ez7f5-ts.js","/_next/static/chunks/3ac71bzdb8krl.js","/_next/static/chunks/2st4jg72-lc-t.js","/_next/static/chunks/3qss25dcigd49.js","/_next/static/chunks/0wod3pr029w__.js","/_next/static/chunks/0luq3wvp5mo1j.js","/_next/static/chunks/3nal99ptqiuu0.js","/_next/static/chunks/0eh37xdptdijx.js","/_next/static/chunks/03zvd0f1-phhi.js","/_next/static/chunks/1raen7xpdk-3t.js","/_next/static/chunks/1qgibnpd1s4yj.js","/_next/static/chunks/3-3x_d1jt_vz5.js","/_next/static/chunks/3pxd56pjid-ea.js","/_next/static/chunks/2ofzleruxyw3g.js","/_next/static/chunks/1wrt-ta9s4-fa.js"],"Image"] 26:I[194704,["/_next/static/chunks/2lo1egizl1har.js","/_next/static/chunks/20-st4r-vmt_9.js","/_next/static/chunks/17wissy38xpia.js","/_next/static/chunks/2lra5r_l71ju3.js","/_next/static/chunks/15o6_p55g4jlt.js","/_next/static/chunks/0xr0_zqqxl0i7.js","/_next/static/chunks/0zvch8mjkmtgu.js","/_next/static/chunks/2xi_wpk87ncci.js","/_next/static/chunks/290ytcwgjf1v4.js","/_next/static/chunks/09jx489or_t-1.js","/_next/static/chunks/0-qqdfbk04_51.js","/_next/static/chunks/38j1pvyhos24t.js","/_next/static/chunks/3s82z7gnp_gmt.js","/_next/static/chunks/0700tvgdr9fs0.js","/_next/static/chunks/2pzt8__sf3khr.js","/_next/static/chunks/2qmp8ez7f5-ts.js","/_next/static/chunks/3ac71bzdb8krl.js","/_next/static/chunks/2st4jg72-lc-t.js","/_next/static/chunks/3qss25dcigd49.js","/_next/static/chunks/0wod3pr029w__.js","/_next/static/chunks/0luq3wvp5mo1j.js","/_next/static/chunks/3nal99ptqiuu0.js","/_next/static/chunks/0eh37xdptdijx.js","/_next/static/chunks/03zvd0f1-phhi.js","/_next/static/chunks/1raen7xpdk-3t.js","/_next/static/chunks/1qgibnpd1s4yj.js","/_next/static/chunks/3-3x_d1jt_vz5.js","/_next/static/chunks/3pxd56pjid-ea.js","/_next/static/chunks/2ofzleruxyw3g.js","/_next/static/chunks/1wrt-ta9s4-fa.js"],"FooterSubscribeLink"] 27:I[241148,["/_next/static/chunks/2lo1egizl1har.js","/_next/static/chunks/20-st4r-vmt_9.js","/_next/static/chunks/17wissy38xpia.js","/_next/static/chunks/2lra5r_l71ju3.js","/_next/static/chunks/15o6_p55g4jlt.js","/_next/static/chunks/0xr0_zqqxl0i7.js","/_next/static/chunks/0zvch8mjkmtgu.js","/_next/static/chunks/2xi_wpk87ncci.js","/_next/static/chunks/290ytcwgjf1v4.js","/_next/static/chunks/09jx489or_t-1.js","/_next/static/chunks/0-qqdfbk04_51.js","/_next/static/chunks/38j1pvyhos24t.js","/_next/static/chunks/3s82z7gnp_gmt.js","/_next/static/chunks/0700tvgdr9fs0.js","/_next/static/chunks/2pzt8__sf3khr.js","/_next/static/chunks/2qmp8ez7f5-ts.js","/_next/static/chunks/3ac71bzdb8krl.js","/_next/static/chunks/2st4jg72-lc-t.js","/_next/static/chunks/3qss25dcigd49.js","/_next/static/chunks/0wod3pr029w__.js","/_next/static/chunks/0luq3wvp5mo1j.js","/_next/static/chunks/3nal99ptqiuu0.js","/_next/static/chunks/0eh37xdptdijx.js","/_next/static/chunks/03zvd0f1-phhi.js","/_next/static/chunks/1raen7xpdk-3t.js","/_next/static/chunks/1qgibnpd1s4yj.js","/_next/static/chunks/3-3x_d1jt_vz5.js","/_next/static/chunks/3pxd56pjid-ea.js","/_next/static/chunks/2ofzleruxyw3g.js","/_next/static/chunks/1wrt-ta9s4-fa.js"],"FooterNewsletterForm"] 4:["$","html",null,{"lang":"en","className":"inter_2bf78b31-module__5Uj77G__variable lora_892e8bd8-module__sLDzVa__variable","suppressHydrationWarning":true,"children":[["$","head",null,{"children":[["$","meta",null,{"name":"robots","content":"max-image-preview:large"}],["$","link",null,{"rel":"preconnect","href":"https://assets.tangente-mag.com"}],["$","link",null,{"rel":"dns-prefetch","href":"https://app.poleditions.com"}]]}],["$","body",null,{"suppressHydrationWarning":true,"children":["$","$L21",null,{"locale":"en","messages":{"common":{"save":"Save","saving":"Saving...","create":"Create","new":"New","delete":"Delete","deleting":"Deleting...","cancel":"Cancel","change":"Change","current":"Current","reset":"Reset","add":"Add","edit":"Edit","copy":"Copy","apply":"Apply","discard":"Discard","back":"Back","loading":"Loading...","untitled":"Untitled","search":"Search...","yes":"Yes","no":"No","from":"From","to":"To","empty":"Empty","unknown":"Unknown","home":"Home","info":"Info","preview":"Preview","view":"View","viewPublic":"View public page","review":"Review","invite":"Invite","pending":"Pending...","noResults":"No results.","today":"Today","properties":"Properties","noResultsFound":"No results found.","year":"Year","month":"Month","day":"Day","week":"Week","createdAt":"Created at","by":"By","published":"Published:","updated":"Updated:","notSet":"Not set","menu":"Menu","openMenu":"Open menu","closeMenu":"Close menu","opensInNewTab":"opens in a new tab","close":"Close","clear":"Clear","redirecting":"Redirecting","hello":"Hello","creating":"Creating...","reloading":"Reloading...","reload":"Reload","refreshCache":"Refresh Cache","errorLoading":"An error occurred while loading data.","errorPageTitle":"Something went wrong","errorPageMessage":"An unexpected error occurred. Please try again or refresh the page.","retry":"Retry","regenerate":"Regenerate","generate":"Generate","processing":"Processing...","loadMore":"Load more","errorSaving":"An error occurred while saving. Please try again.","characters":"{count}/{max} characters","sync":"Sync","resume":"Resume","copiedToClipboard":"Copied to clipboard","copied":"Copied","notFound":"Not found","error":"An error occurred. Please try again.","saved":"Saved successfully","actionCannotBeUndone":"This action cannot be undone.","folder":"Folder","confirm":"Confirm","remove":"Remove","previous":"Previous","next":"Next","actions":"Actions","uploading":"Uploading…","all":"All","confirmDelete":"Delete?","skip":"Skip","toggle":"Toggle","mathFraction":"Fraction","mathExponent":"Exponent","mathIndex":"Subscript","mathSquareRoot":"Square root","mathPi":"Pi","mathLessOrEqual":"Less than or equal","mathGreaterOrEqual":"Greater than or equal","mathNotEqual":"Not equal","mathMultiply":"Multiply","mathInfinity":"Infinity","mathSum":"Sum","mathIntegral":"Integral","insertFormula":"Insert a formula","composeFormula":"Compose your formula","loadingEditor":"Loading the editor…","insert":"Insert","pagination":"Pagination","previousPage":"Previous page","nextPage":"Next page","alphabetNavAriaLabel":"Alphabetical navigation","showAll":"All","skipToContent":"Skip to main content","selectMonth":"Select a month..."},"status":{"published":"Published","draft":"Draft","active":"Active","inactive":"Inactive","public":"Public","private":"Private","verified":"Verified","subscribers":"Subscribers Only","notVerified":"Not verified","canceled":"Canceled","pending":"Pending","toReview":"To Review","reviewed":"Reviewed","accepted":"Active","suspended":"Suspended"},"nav":{"overview":"Overview","dashboard":"Dashboard","statistics":"Statistics","subscriptionCockpit":"Subscription Cockpit","executiveDashboard":"Dashboard","content":"Content","magazines":"Magazines","issues":"Issues","planning":"Planning","folders":"Folders","articles":"Articles","categories":"Categories","enums":"Enums","articleTypes":"Article Types","magazineSettings":"Article AI","customers":"Customers","customerGroups":"Customer Groups","users":"Users","organizations":"Organizations","subscriptions":"Subscriptions","invoices":"Invoices","prelevement":"SEPA Direct Debit","productsAndPlans":"Products & Plans","plans":"Plans","products":"Products","prices":"Prices","features":"Features","coupons":"Coupons","engagement":"Engagement","notifications":"Notifications","administration":"Administration","admins":"Admins","etablissements":"Schools","appSettings":"App Settings","routingFiles":"Routing Files","messengerMonitor":"Messenger Monitor","stripeWebhookEvents":"Stripe webhooks","system":"System","maintenance":"Maintenance","editorial":"Editorial","podcasts":"Podcasts","podcastEpisodes":"Episodes","podcastPlaylists":"Playlists","podcastFigures":"Figures","podcastDomaines":"Domains","taxonomy":"Taxonomy","billing":"Billing","catalog":"Catalog","aiAutomation":{"value":"AI & Automation","magazine":"Magazine","comments":"Comments"},"aiKeys":"AI API Keys","dev":"Dev","subscribers":"Subscribers","marketing":"Marketing","commerce":"Commerce","translations":"Translations","reviewQueue":"Review queue","translationSettings":"Translation settings","uiCatalog":"Interface text","authors":"Authors","portraitConsents":"Portrait consent","shopifyLibrary":"Infinimath library","shopifyProducts":"Products","shopifySettings":"Shopify settings","shopifyAiEnrichment":"AI enrichment","organisation":"Organization","problems":{"value":"Problems","aiConfig":"AI configuration","import":"Import"},"public":{"featured":"Featured","easyMath":"Easy Math","themes":"Themes","articles":"All articles","historyCulture":"History & Culture","gamesChallenges":"Games & Challenges","magazines":"Magazines","folders":"Folders","tangenteAudio":"Tangente Audio","tangenteJeux":"Tangente Jeux","mainNavigation":"Main navigation","mobileNavigation":"Navigation menu","videos":"Videos","readingNotes":"Reading Notes","mathsForAll":"Math for All","curiosity":"Curiosity","dailyMath":"Everyday Math","mathForEveryone":"Math for Everyone","knowledge":"Knowledge","interview":"Interview","famousPersonalities":"Famous Personalities","mathAndHistory":"Math & History","mathAndPhilosophy":"Math & Philosophy","mathAndArt":"Math & Art","problems":"Problems","shortStory":"Short Story","recremaths":"Récrémath","myFavoriteProblems":"My Favorite Problems","amazingMath":"Amazing Math","logicMatters":"Logic Matters","mathThemes":"Mathematical Themes","geometry":"Geometry","algebra":"Algebra","analysis":"Analysis","arithmetic":"Arithmetic","numerical":"Numerical","logic":"Logic","combinatoricsAndGames":"Combinatorics & Games","appliedMathematics":"Applied Mathematics","probabilityAndStatistics":"Probability & Statistics","teacherArea":"Teacher Area","mathOlympiads":"Math Olympiads","contact":"Contact"},"diffusionLists":"Diffusion Lists","supportRequests":"Support Tickets","newsroom":{"value":"NewsRoom","dashboard":"Dashboard","sources":"Sources","feed":"Feed","selections":"Selections","preDrafts":"Pre-drafts","settings":"Settings"},"magazineImports":"Mineru imports","magazineImportSettings":"Import settings","newsletter":"Newsletter","contacts":"Contacts","contactsAll":"All Contacts","contactsConsents":"Consents","emailLog":"Email Log","contactSegments":"Segments","campaigns":"Campaigns","campaignTemplates":"Email Templates","emailAutomations":"Automations","emailAutomationSend":"Manual Send","games":"Games","giftGrants":"Gifts to fulfil","giftPackages":"Gift offers","commentModeration":"Comment moderation","crm":"Anti-churn","shopifySubscriptionOffers":"Bookshop subscription offers","shopifySubscriptionOfferSettings":"Bookshop offer settings","shopifyReceiptClaims":"Receipt claims"},"table":{"rowsPerPage":"Rows per page","pageOf":"Page {page} of {total}","rowsSelected":"{count} of {total} row(s) selected.","goToFirstPage":"Go to first page","goToPreviousPage":"Go to previous page","goToNextPage":"Go to next page","goToLastPage":"Go to last page","selectAll":"Select all","selectRow":"Select row","toggleColumns":"Toggle columns","clearFilters":"Clear filters","selected":"{count} selected","failedToLoad":"Failed to load data. Please try again.","retry":"Retry","asc":"Asc","desc":"Desc","hide":"Hide","collapseRow":"Collapse","expandRow":"Expand","moreRows":"{count} more"},"field":{"name":"Name","slug":"Slug","email":"Email","title":"Title","description":"Description","summary":"Summary","content":"Content","cover":"Cover","logo":"Logo","image":"Image","active":"Active","sequence":"Display order","sequenceDescription":"Controls the display priority of articles with this type. Default is 0. Use -1 to show last, positive values to show first.","upcoming":"Upcoming","upcomingDescription":"Shows the issue in the public grid with its cover only and its release date. Mutually exclusive with \"Published\".","published":"Published","visibility":"Visibility","featured":"Featured","featuredImage":"Featured Image","coverImage":"Cover Image","categories":"Categories","keywords":"Keywords","metadata":"Metadata","invalidJson":"Invalid JSON.","invalidJsonObject":"JSON must be an object.","articleType":"Article Type","author":"Author","brand":"Brand","shopifyProduct":"Shopify Product","shopifyProducts":"Shopify Products","brandColor":"Brand Color","coverRatio":"Cover Ratio","issueNumber":"Issue Number","publishedAt":"Published At","parentCategory":"Parent Category","noParent":"No Parent","label":"Label","key":"Key","type":"Type","values":"Values","status":"Status","locale":"Language","duration":"Duration","comments":"Comments","fields":"Fields","parent":"Parent","children":"Children","role":"Role","joined":"Joined","actions":"Actions","member":"Member","language":"Language","dateFormat":"Date format","phone":"Phone","refId":"Ref ID","subscription":"Subscription","plan":"Plan","balance":"Balance","created":"Created","registeredDate":"Registered Date","phoneNumber":"Phone Number","customFields":"Custom Fields","firstName":"First Name","lastName":"Last Name","address":"Address","addressLine1":"Address Line 1","city":"City","postalCode":"Postal Code","country":"Country","createdAt":"Created At","updatedAt":"Updated At","color":"Color","manual":"Manual","placeholder":{"searchByTitle":"Search by title...","searchByName":"Search by name...","enterArticleTitle":"Enter article title...","articleDescription":"Write a brief summary of the article...","briefDescription":"Brief description of the article...","selectVisibility":"Select visibility...","selectArticleType":"Select article type...","selectCategories":"Select categories...","selectCategory":"Select category...","selectBrand":"Select brand...","selectFolder":"Select folder...","selectEnum":"Select enum...","selectType":"Select type...","selectDynamic":"Select {label}...","addDynamic":"Add {label}...","selectRatio":"Select ratio","coverRatio":"e.g., 16:9, 1:1.414","fieldKey":"field_key","enumLabel":"e.g., Pending","enterNumber":"Enter a number...","selectCountry":"Select a country","searchCountry":"Search...","noCountryFound":"No country found.","selectLanguage":"Select language...","searchLanguage":"Search language...","noLanguageFound":"No language found.","select":"Select...","search":"Search..."},"translations":"Translations","commentsEnabled":"Comments open","commentsVisible":"Comments shown","commentsUnavailableForType":"Problème articles take no comments: the thread would walk past the hidden solution."},"message":{"success":{"created":"{entity} created successfully","updated":"{entity} updated successfully","deleted":"{entity} deleted successfully"},"error":{"deleted":"Failed to delete {entity}","notFound":"{entity} not found","failedToLoad":"Failed to load data. Please try again.","generic":"Something went wrong. Please try again."},"confirm":{"delete":"Are you sure you want to delete this {entity}?","deleteDescription":"This action cannot be undone. This will permanently delete the selected item and all associated data.","removeFolderFromIssue":"Are you sure you want to remove this folder from the issue? The folder and its articles will no longer be associated with this issue.","removeFromFolder":"Are you sure you want to remove this article from the folder?","changeArticleType":"Change article type?","changeArticleTypeDescription":"Changing the article type may reset some custom field values. This cannot be undone."}},"podcast":{"listen":"Listen to the Podcast","listenShort":"Podcast","paywallTitle":"Keep listening","paywallDescription":"Subscribe to listen to the full podcast of this issue.","paywallArticleDescription":"Subscribe to listen to the full podcast of this article.","popup":{"eyebrow":{"podcast":"PODCAST","tag":"DEBATE"},"listenLabel":"Listen to the podcast","play":"Play","pause":"Pause","close":"Close","originalLabel":"An original Tangente podcast","subscribeCta":"Subscribe to listen in full","previewNote":"Free preview. The full episode is for subscribers only.","presentersAlt":"The two hosts of the Tangente podcast","scrub":"Seek through the episode","unavailable":"The episode is being prepared."}},"users":{"welcome":"Welcome","generalInformation":"General Information","profilePhoto":"Profile Photo","handle":"Username","handleChangeOnce":"You can change your username once. Choose carefully.","handleRules":"3 to 24 characters: letters, digits, hyphens and underscores.","handleChecking":"Checking availability…","handleAvailable":"This username is available.","handleTaken":"This username is already taken.","handleSuggestions":"Available instead:","handleAlreadyChanged":"You have already used your one username change.","handleConfirmTitle":"Change your username?","handleConfirmDescription":"Your username will become {handle}. You get only one change, and it cannot be undone.","handleConfirmAction":"Change my username","firstName":"First Name","firstNamePlaceholder":"Your first name","lastName":"Last Name","lastNamePlaceholder":"Your last name","changeEmail":"If needed, change your email below, and we will send you a verification link.","password":"Password","changePasswordDescription":"To change your password please fill the current one.","currentPassword":"Current password","newPassword":"New password","repeatNewPassword":"Repeat new password","displayPreferences":"Display preferences","notifications":"Notifications","productUpdates":"Product updates","newsAndAnnouncements":"News and announcements","deleteAccount":"Delete Account","deleteAccountWarning":"This action will:","removePersonalData":"Remove all your personal data","cannotBeUndone":"Cannot be undone","typeDeleteToConfirm":"Type \"DELETE\" to confirm:","deleting":"Deleting...","continue":"Continue","connectedAccounts":"Connected accounts","connectedAccountsDescription":"Manage third-party services you use to log in to your Tinct account","ssoInfo":"You can connect multiple accounts. At least one account must remain connected to access Tinct. Otherwise, opt for the email - password option and set a password below.","google":"Google","linkedin":"LinkedIn","microsoft":"Microsoft","english":"English","french":"French","spanish":"Spanish","german":"German","actions":{"editUser":"Edit User","deleteUser":"Delete User","banUser":"Ban User","unbanUser":"Unban User","sendResetPassword":"Send Reset Password Link","impersonate":"Impersonate","impersonateSuccess":"Now impersonating {email}","impersonateError":"Failed to impersonate user"},"deleteTitle":"Delete user","deleteDescription":"This action cannot be undone. The user and linked customer will be deleted or archived depending on their existing data.","deleteSuccess":"User deleted successfully","deleteArchived":"The user has been archived because they have historical data.","deleteBlocked":"Cannot delete this user because they have an active subscription. Please cancel the subscription first.","deleteAdminForbidden":"Cannot delete an administrator account.","deleteError":"Failed to delete user","deleteWarning":"This action is irreversible. The user and all associated data will be permanently deleted.","banSuccess":"User has been banned","banError":"Failed to ban user","banWarning":"This user will be suspended and will no longer be able to access the platform.","unbanSuccess":"User has been unbanned","unbanError":"Failed to unban user","unbanWarning":"This user will be reactivated and will regain access to the platform.","resetPasswordSent":"Reset password link sent","resetPasswordError":"Failed to send reset password link","resetPasswordDialog":{"action":"Reset password","title":"Reset password","description":"Send the password reset email to {email}, or generate a link to pass on. The account holder chooses the new password.","descriptionNoEmail":"Send the password reset email to the account holder, or generate a link to pass on. The account holder chooses the new password.","themeLabel":"Reset page branding","theme":{"default":"Tangente","podcast":"Tangente Podcast","jeux":"Tangente Jeux"},"send":"Send email","generate":"Generate link","url":"Link to share","copy":"Copy link","copyError":"Unable to copy the link. Select it and copy it manually.","expiry":"This link expires in {minutes} minutes and works once. Only share it with the account holder: it lets whoever holds it set a new password and sign in.","sent":"Reset email sent","error":{"notMintable":"No link can be generated for a staff account (administrator or permission holder). Send the email instead.","rateLimited":"Too many requests in the last hour. Please try again later."}},"editPageTitle":"User Details","email":"Email","roles":"Roles","noRoles":"No roles","viewStripeCustomer":"View Stripe Customer","createStripeCustomer":"Create Stripe Customer","createStripeCustomerSuccess":"Stripe customer created successfully","createStripeCustomerError":"Failed to create Stripe customer"},"customers":{"title":"Customers","new":"New Customer","search":"Search customers...","searchByName":"Name","searchByNamePlaceholder":"Search by name...","searchByEmail":"Email","searchByEmailPlaceholder":"Search by email...","searchByCustomerNumber":"Customer #","searchByCustomerNumberPlaceholder":"Search by number...","searchByPostalCode":"Postal code","searchByPostalCodePlaceholder":"Search by postal code...","searchByTown":"Town","searchByTownPlaceholder":"Search by town...","filterOrigin":"Origin","filterManualCreated":"Manually created","filterTerm":"Subscription term","termMonthly":"Monthly","termSemester":"Semi-annual","termAnnual":"Annual","termBiennial":"2 years","filterSubscriptionStart":"Subscription start","edit":"Edit Customer","delete":"Delete Customer","subscriptions":{"value":"Subscriptions","title":"Subscriptions","description":"Customer subscription plans.","noSubscriptions":"No subscriptions found.","upcomingTitle":"Upcoming subscription","upcomingDescription":"The customer has requested a plan change. The new subscription is pending and will take effect at the end of the current billing period.","plan":"Plan","periodStart":"Period Start","periodEnd":"Period End","statusActive":"Active","statusTrialing":"Trialing","statusPastDue":"Past Due","statusCanceled":"Canceled","statusUnpaid":"Unpaid","statusIncomplete":"Incomplete","statusExpired":"Expired","statusPaused":"Paused","alreadyRenewed":"Already Renewed","editSubscriptions":"Manage Customer Subscription","viewSubscription":"View subscription","renewSubscription":"Renew Subscription","subscriptionRenewed":"Subscription renewed","renewError":"Failed to renew subscription","renewConfirmTitle":"Renew subscription?","renewConfirmDescription":"This will renew the {plan} subscription. The current period ends on {endDate}.","createSubscription":"Create Subscription","createSubscriptionDescription":"Create a new subscription for this customer.","selectPrices":"Select prices","paymentMethod":"Payment method","paymentMethodCard":"Card","paymentMethodCheque":"Cheque","paymentMethodVirement":"Bank transfer","paymentMethodPrelevement":"Direct debit","paymentMethodBon":"Bon","paymentMethodPrelevementSepa":"SEPA direct debit (batch)","subscriptionCreated":"Subscription created","createError":"Failed to create subscription","missingDeliveryAddress":"This customer has no delivery address. Please add a delivery address before creating a paper subscription.","horsMetropoleBlocked":"This customer's delivery address is outside metropolitan France. Please enable the \"Hors Métropole\" option or update the customer's delivery address.","metropoleWithHorsMetro":"This customer's delivery address is in metropolitan France. The \"Hors Métropole\" option is not compatible with a metropolitan address.","viewOnStripe":"View on Stripe","changeSubscription":"Change Subscription","changeSubscriptionDescription":"Change the subscription plan for this customer.","subscriptionChanged":"Subscription changed","changeError":"Failed to change subscription","selectNewPlan":"Select a new plan","noActiveSubscription":"No active subscription","noPricesSelected":"Please select at least one price","selectPlan":"Select a plan","duration":"Duration","addons":"Add-ons","activateNow":"Activate subscription now","activateNowHint":"By default, Stripe marks the subscription as incomplete pending payment. Check this to activate the subscription immediately and assign quotas.","activateNowRenewHint":"By default, quotas are only assigned after the invoice is paid. Check this to assign quotas for the new period immediately.","oneTimeSubscription":"One-time subscription (no renewal)","oneTimeSubscriptionHint":"The subscription will be automatically canceled at the end of the current period and will not renew."},"untitled":"Untitled","notFound":"Customer not found","info":{"title":"Customer Information","description":"General customer details.","billingAddress":"Billing Address","billingAddressDescription":"Customer billing address.","deliveryAddress":"Delivery Address","deliveryAddressDescription":"Customer delivery/shipping address.","differentBillingAddress":"Different billing address","firstName":"First Name","lastName":"Last Name","addressLine1":"Address Line 1","addressLine2":"Address Line 2","city":"City","postalCode":"Postal Code","country":"Country","currency":"Currency","customerType":"Customer Type","civility":"Civility","siret":"SIRET","agencyBilling":"Agency billing","agencyBillingDescription":"Link this customer to an agency so that invoices and billing addresses are managed by the agency.","agencyLinked":"Invoices will be sent to this agency","searchAgency":"Search for an agency...","searchAgencyPlaceholder":"Type agency name or email","noAgencyFound":"No agency found","agencyEstablishments":"Linked establishments","agencyEstablishmentsDescription":"Customers linked to this agency.","noEstablishments":"No establishments linked to this agency."},"customerType":{"individual":"Individual","establishment":"Establishment","professor":"Professor","student":"Student","researcher":"Researcher","library":"Library","retired":"Retired","engineer":"Engineer","documentalist":"Documentalist","inspector":"Inspector","agency":"Agency"},"civility":{"groupPerson":"Person","groupInstitution":"Institution","mr":"Mr","mrs":"Mrs","ms":"Ms","mx":"Mx","dr":"Dr","college":"College","lycee":"High School","university":"University","school":"School","library":"Library","mediatheque":"Media Library","association":"Association","institution":"Institution"},"additionalInfo":{"title":"Additional Information","description":"Extra metadata for this customer."},"notifications":{"title":"Notifications","description":"Control emails sent to this customer.","invoiceEmail":"Invoice email","invoiceEmailDescription":"Send an email when an invoice is created."},"quotas":{"title":"Quotas","description":"Customer usage quotas.","noQuotas":"No quotas available.","updateUsage":"Update usage","updateUsageDescription":"Adjust quota usage for this customer.","featureKey":"Feature","amount":"Amount","operation":"Operation","operationAdd":"Add","operationConsume":"Consume","operationRemove":"Remove","description_field":"Description","issueRange":"Issue range","periodStart":"Period start","periodEnd":"Period end","usageUpdated":"Quota updated","updateError":"Failed to update quota","updateIssueNumbers":"Update issue numbers","updateIssueNumbersDescription":"Update the issue number range for this quota.","issueNumberStart":"Start number","issueNumberEnd":"End number","issues":"issues","issueNumbersUpdated":"Issue numbers updated","issueNumbersUpdateError":"Failed to update issue numbers","createQuota":"Create Quota","createQuotaDescription":"Manually create a quota for a feature. Only features without an active quota are shown.","selectFeature":"Select a feature","noAvailableFeatures":"All features already have an active quota.","quotaLimit":"Quota limit","quotaCreated":"Quota created successfully","createError":"Failed to create quota","manualQuotaCreation":"Manual quota creation","duration":"Duration","duration1Month":"1 month","duration2Months":"2 months","duration3Months":"3 months","duration6Months":"6 months","duration1Year":"1 year","duration2Years":"2 years","updateDigitalAccess":"Update digital access","accessEnabled":"Access enabled","periodEndMustBeAfterStart":"End date must be after start date"},"invoices":{"title":"Invoices","description":"Customer invoice history.","noInvoices":"No invoices available.","date":"Date","invoiceDate":"Invoice date","number":"Number","amount":"Amount","periodStart":"Period start","periodEnd":"Period end","download":"Download","downloadError":"Unable to download invoice.","viewOnStripe":"View on Stripe","sync":"Sync invoice","syncSuccess":"Invoice synced","syncError":"Failed to sync invoice","statusPaid":"Paid","statusOpen":"Open","statusDraft":"Draft","statusVoid":"Void","statusUncollectible":"Non paid","statusFailed":"Payment failed","pay":"Pay","payConfirmTitle":"Pay this invoice?","payConfirmDescription":"This will mark the invoice as paid.","paySuccess":"Invoice paid","payError":"Failed to pay invoice","payReferenceLabel":{"card":"Last 4 digits of card","cheque":"Cheque number","virement":"Transfer reference","prelevement":"Direct debit reference","bon":"Bon number"},"payReferencePlaceholder":{"card":"1234","cheque":"0001234","virement":"TRF-2026-001","prelevement":"DD-2026-001","bon":"BON-001"},"payReferenceRequired":"This field is required","earlyRenewal":"Early Renewal","needsVoid":"Needs void","void":"Void invoice","voidSuccess":"Invoice voided","voidError":"Failed to void invoice","send":"Send to customer","sendSuccess":"Invoice sent to customer","sendError":"Failed to send invoice","viewOnStripeDashboard":"Stripe Dashboard","linkPennylane":"Link Pennylane invoice","pennylaneTitle":"Pennylane Invoice","pennylaneLabel":"Pennylane invoice number","pennylanePlaceholder":"e.g. PL-2026-001","pennylaneSuccess":"Pennylane number updated","pennylaneError":"Failed to update Pennylane number","viewDetails":"View details","detailsTitle":"Invoice details","invoiceId":"Invoice ID","amountPaid":"Amount paid","amountRemaining":"Amount remaining","customerEmail":"Customer email","dueDate":"Due date","paidAt":"Paid on","paymentReference":"Payment reference","lineItems":"Line items","paymentMethodStripe":"Stripe (online payment)","payNow":"Pay now","notPayable":"This invoice cannot be paid online. Please contact customer support.","resubscribe":"Resubscribe","paymentLinkError":"Unable to retrieve the payment link.","canceledSubscriptionNotice":"This subscription has been canceled. Redirecting you to resubscribe…","resubscribeUnavailable":"We can't resume this subscription automatically. Choose a plan to continue.","resubscribeError":"Unable to start the resubscribe checkout. Please try again.","accountDescription":"Your invoice history.","statusToPay":"Due"},"quotaTransactions":{"title":"Quota History","description":"Customer quota transaction tracking.","noTransactions":"No transactions found.","allOperations":"All operations","date":"Date","feature":"Feature","operation":"Operation","amount":"Amount","descriptionColumn":"Description","operationAllocate":"Allocate","operationConsume":"Consume","operationAdjust":"Adjust","operationRefund":"Refund","operationReset":"Reset","pageInfo":"Page {current} of {total}"},"saveSuccess":"Customer updated successfully","addressCountryRestriction":"The selected country is not compatible with your current subscription. Your subscription does not include a shipping option outside metropolitan France. Please contact support if you need help modifying your subscription.","createTitle":"New Customer","createSuccess":"Customer created successfully","deleteTitle":"Delete customer","deleteDescription":"This action cannot be undone. The customer and linked user will be deleted or archived depending on their existing data.","deleteBlocked":"Cannot delete this customer because they have an active subscription. Please cancel the subscription first.","deleteArchived":"The customer has been archived because they have historical data (subscriptions, invoices).","deleteSuccess":"Customer deleted successfully.","deleteAdminForbidden":"Cannot delete an administrator account.","validation":{"nameRequired":"Name is required","invalidEmail":"Invalid email format"},"tabs":{"info":"Info","addresses":"Addresses","subscriptions":"Subscriptions","subscription":"Subscription","billing":"Billing","usage":"Usage","activity":"Activity","emails":"Emails"},"emails":{"title":"Emails sent","description":"History of emails sent to {email} — campaigns and transactional.","noEmailAddress":"This customer has no email address on file.","columns":{"date":"Sent on","kind":"Type","subject":"Subject / Template","reference":"Reference"},"kind":{"campaign":"Campaign","transactional":"Transactional"}},"summary":{"customerNumber":"Customer #","title":"Summary","subscription":"Subscription","balance":"Balance","delinquent":"Delinquent","createdAt":"Created","updatedAt":"Updated","active":"Active","inactive":"Inactive","userStatus":"User status","resendInvitation":"Resend invitation","invitationResent":"Invitation resent successfully","invitationResendError":"Failed to resend invitation"},"linkedUser":"Linked user","viewUser":"View user","convertToEtablissementAction":"Convert to etablissement","convertToEtablissementTitle":"Convert this customer into an etablissement","convertToEtablissementDescription":"Creates an etablissement record with this customer as CDI. If the customer already has an active paper subscription, the etablissement will be activated immediately and seats opened. Digital-only subscriptions do not grant seats.","convertToEtablissementNameLabel":"Etablissement name","convertToEtablissementConfirm":"Create etablissement","convertToEtablissementSuccess":"Etablissement created.","convertToEtablissementAlreadyExists":"This customer is already linked to an etablissement. Redirecting.","convertToEtablissementNoUser":"This customer is not linked to a user; cannot create an etablissement.","linkToEtablissementAction":"Link to an etablissement","linkToEtablissementTitle":"Link this customer to an existing etablissement","linkToEtablissementDescription":"Attaches this customer as the CDI of an etablissement that has none. If the customer has an active paper subscription, the etablissement becomes ACTIVE and seats are issued immediately.","linkToEtablissementSelectLabel":"Etablissement to link","linkToEtablissementPlaceholder":"Pick an etablissement…","linkToEtablissementEmpty":"No CDI-less etablissement available.","linkToEtablissementConfirm":"Link as CDI","linkToEtablissementSuccess":"Customer linked to the etablissement.","linkToEtablissementAlreadyLinked":"This customer is already CDI of an etablissement, or the chosen etablissement already has a CDI.","linkToEtablissementNoUser":"This customer is not linked to a user; cannot use it as a CDI.","linkToEtablissementAlreadyMember":"This user is already a member of this etablissement (likely the professor who submitted the form). Pick another customer.","classification":{"title":"Classification","description":"Assign this customer to one or more groups."},"paymentMethod":{"title":"Payment method","description":"Usable payment method on file for this customer, preferring the default one.","method":"Method","none":"No payment method","card":"Card","sepa":"SEPA Direct Debit","link":"Stripe Link","other":"Other payment method","expiry":"Expiry","expired":"Expired","sponsoredBy":"Paid by (gifted subscription)"},"bankInfo":{"title":"Bank Information","description":"Direct debit details (read-only, imported data).","editTitle":"Edit Bank Information","editDescription":"Update the IBAN, BIC, and SEPA mandate details.","ibanPlaceholder":"Enter the new full IBAN","ibanHint":"Leave blank to keep the current IBAN.","updateSuccess":"Bank information updated","updateError":"Failed to update bank information","noData":"No SEPA direct debit configured for this customer.","create":"Add SEPA direct debit","amount":"Amount","subscription":"Subscription","status":"Status","signatureDate":"Signature date","debitDate":"Debit date","suspensionDate":"Suspension date","mandateReference":"Mandate reference"}},"comments":{"title":"Comments","description":"Internal notes and comments about this customer.","placeholder":"Write a comment...","send":"Send","empty":"No comments yet.","unknownAuthor":"Unknown","moderation":{"title":"Comment moderation","description":"Reader comments awaiting a decision. Approving publishes the comment and rebuilds the article page.","column":{"author":"Author","comment":"Comment","article":"Article","status":"Status","spamScore":"Spam score","createdAt":"Posted on"},"status":{"pending":"Pending","approved":"Approved","rejected":"Rejected","spam":"Spam","needs_review":"Needs review","low_value":"No content"},"bulk":{"approve":"Approve","reject":{"value":"Reject","confirmTitle":"Reject these comments?","confirmDescription":"Rejected comments disappear from the article. Replies to a root comment are rejected with it."},"spam":{"value":"Mark as spam","confirmTitle":"Mark these comments as spam?","confirmDescription":"Their author still sees them as pending. Replies to a root comment follow it."},"restore":"Send back to pending"},"applied":"{count, plural, one {# comment updated} other {# comments updated}}","skipped":"{count, plural, one {# comment skipped: its status had changed} other {# comments skipped: their status had changed}}","accessDenied":"You do not have access to this page.","detail":{"action":"View comment","title":"Comment","description":"The submitted text, its context and the last moderation decision.","spamReasons":"Spam signals","moderatedBy":"Moderated by","moderatedAt":"Moderated on","reason":"Reason","inReplyTo":"In reply to @{handle}","parentDeleted":"deleted","aiScore":"AI score","aiCategories":"AI categories","aiReason":"Why the AI scored it that way","aiUnavailable":"Classification unavailable"},"viewOnSite":"View on the site","reason":{"off_topic":"Off topic","abusive":"Abusive","spam_links":"Unwanted links","personal_data":"Personal data","duplicate":"Duplicate","low_value":"Greeting only","unclear_purpose":"Unclear link purpose","other":"Other"},"reclassify":"Re-analyse with AI","reclassifyQueued":"Queued for re-analysis. The verdict lands shortly."},"settings":{"title":"Comment moderation","save":"Save settings","saved":"Settings saved.","saveFailed":"Could not save settings.","loadFailed":"Could not load settings","loadFailedHint":"Reload the page. If it persists, the comment settings API is unreachable.","discardChanges":"Discard changes","general":{"title":"Comments","description":"Whether readers can post, and where a new comment lands before anyone reads it."},"enabled":"Accept comments","enabledHint":"Turning this off closes new submissions. Existing discussions stay visible.","moderation":"Hold comments for review","moderationHint":"On, a new comment waits in the queue. Off, it publishes immediately and the AI can still take it down. The spam gate runs either way.","trustAfterApproved":"Publish without review after N approved comments","trustAfterApprovedHint":"0 trusts nobody. A trusted author still passes the spam gate every time.","spam":{"title":"Spam gate","description":"Local checks that run on every comment, before anything else. A comment they catch is never sent to the classifier."},"maxLinks":"Links allowed per comment","blockedTerms":"Blocked terms","blockedTermsHint":"One term per line. Matching is case-insensitive.","ai":{"title":"AI classification","description":"A model grades each comment the spam gate cleared, off the request path: the comment publishes or waits per the rules above, then the verdict moves it.","enabled":"Classify comments with AI","mode":"Who gets classified","modeConditional":"Everyone except trusted authors","modeStrict":"Every comment, trusted or not","provider":"Provider","model":"Model","autoApproveBelow":"Publish automatically below","autoApproveBelowHint":"0 means the classifier never publishes on its own — it can only hold or reject.","autoRejectAtOrAbove":"Reject automatically at or above","bandHint":"Anything between the two bounds goes to the moderation queue.","rejectCategories":"Reject outright on these categories","rejectCategoriesHint":"A hit here rejects whatever the score says. Spam is filed as spam rather than rejected.","category":{"hate":"Hate","harassment":"Harassment","sexual":"Sexual content","violence":"Violence","self_harm":"Self-harm","spam":"Spam","off_topic":"Off topic","unconstructive":"No argument given"},"temperature":"Temperature","maxTokens":"Max tokens","timeoutSeconds":"Timeout (s)","systemPrompt":"Grading instructions","systemPromptHint":"What counts as a violation here. The answer format and the rule that a comment is data, never an instruction, are fixed and cannot be edited."}}},"subscriptions":{"title":"Subscriptions","searchPlaceholder":"Search subscriptions...","details":"Subscription Details","detailsDescription":"View and manage this subscription.","subscriptionId":"Subscription ID","currentPeriodStart":"Current Period Start","currentPeriodEnd":"Current Period End","periodStart":"Period Start","periodEnd":"Period End","canceledAt":"Canceled At","canceling":"Canceling","cancelingAtPeriodEnd":"Canceling at Period End","cancelAtPeriodEnd":"Cancel at Period End","cancelNow":"Cancel Now","cancelNowConfirm":"Cancel Subscription","cancelNowDescription":"Are you sure you want to cancel this subscription immediately?","viewSubscription":"View Subscription","plan":"Plan","user":"User","items":"Items","metadata":"Metadata","noMetadata":"No metadata available.","syncSuccess":"Subscription synced from Stripe","syncError":"Failed to sync subscription","syncToStripe":"Sync to Stripe","syncToStripeSuccess":"Subscription synced to Stripe","syncToStripeError":"Failed to sync subscription to Stripe","cancelAtPeriodEndConfirm":"Cancel at period end?","cancelAtPeriodEndDescription":"The subscription will remain active until the end of the current billing period, then be canceled.","cancelAtPeriodEndSuccess":"Subscription will be canceled at period end","cancelError":"Failed to cancel subscription","cancelNowSuccess":"Subscription canceled immediately","resumeSuccess":"Subscription resumed","resumeError":"Failed to resume subscription","notFound":"Subscription not found","statusActive":"Active","statusTrialing":"Trialing","statusPastDue":"Past Due","statusCanceled":"Canceled","statusUnpaid":"Unpaid","statusIncomplete":"Incomplete","statusExpired":"Expired","statusPaused":"Paused","statusCancelPending":"Cancel pending","periodStartMonth":"Start Period Month","periodEndMonth":"End Period Month"},"cms":{"openEditor":"Open Editor","noContentYet":"No content yet","blockCount":"{count} block","blockCountPlural":"{count} blocks"},"login":{"redirecting":"Redirecting","loggingOut":"Signing out…","oauthCallback":"OAuth Callback Page","verifyEmail":{"title":"Email verified","description":"Your email has been verified. Sign in on the browser where you started your purchase to continue, or sign in here to start over.","redirecting":"Redirecting...","verifiedRedirecting":"Your email has been verified. Redirecting..."}},"heroTangente":{"topic":{"geometry":"Geometry","algebra":"Algebra","analysis":"Analysis","arithmetic":"Arithmetic","numerical":"Numerical computation","logic":"Logic","combinatorics_and_games":"Combinatorics & Games","applied_mathematics":"Applied Mathematics","probability_and_statistics":"Probability & Statistics"},"topicFallback":"Mathematics","heading":"Tangente, the mathematics magazine","lede":"From living things to the stars, Tangente reveals the mathematical connections shaping our world.","regionLabel":"Tangente hero","destinationsLabel":"Discover Tangente","switcherLabel":"Switch hero slide","slideMagazine":"Magazine","slideGames":"Games","slidePodcast":"Podcast","linkMagazines":"Browse all issues","linkGames":"Play logic games","linkAudio":"Listen to the podcast","slides":{"library":{"titlePrimary":"To better understand","titleSecondary":"the world","lead":"From living things to the stars, Tangente reveals the mathematical connections that shape our world.","leadShort":"The mathematical connections that shape our world.","browseLabel":"Browse the issues","discoverLabel":"Discover subscriptions","factsIssues":"issues","selectionAnnouncement":"Now showing: {label} — {title}","factsCulture":"15 years of mathematical\nculture","factsGames":"Thousands of\ngames and puzzles","factsNews":"Mathematical\nnews","factsEverywhere":"Everywhere\nwith you","companionTitlePrimary":"Curiosity","companionTitleSecondary":"goes with you","companionLead":"Mathematics made accessible at last: nature, science, history, culture and technology go with you everywhere.","companionLeadShort":"Mathematics made accessible at last, everywhere with you.","exploreLabel":"Start exploring","stageLabel":"Tangente collection in three dimensions. Left and right arrows to change issue, Enter to open it, F to switch facet.","unsupportedLabel":"3D rendering is unavailable on this device.","detailLinkLabel":"Read this issue","detailCloseLabel":"Close the summary and return to the library","detailCoverAlt":"Cover — {label}, {title}","detailPages":"{count} pages","detailSummaryFallback":"The contents of this issue will be online soon."},"podcast":{"titlePrimary":"All of Tangente","titleSecondary":"in podcast form","lead":"Our authors' original articles, told through a conversation anyone can follow.","playLabel":"Start listening","discoverLabel":"Discover the platform","pauseLabel":"Pause","quietPlayLabel":"Resume listening","quietPauseLabel":"Pause listening","panelLabel":"The studio","factsEpisodes":"episodes","factsUniverses":"topics","factsNewLead":"New","factsNew":"every week","factsStudioLabel":"The studio in figures","factsSeriesLabel":"The series in figures","seriesPanelLabel":"The series","seriesTitlePrimary":"Warm voices","seriesTitleSecondary":"in a format anyone can follow","seriesLead":"Explore the series and carry on listening at the pace of your own curiosity.","exploreLabel":"Explore the episodes","previewLabel":"Preview ·","stageLabel":"Tangente podcast scene. Left and right arrows to change episode, Enter to start or pause listening.","listenerAlt":"The host of the Tangente podcast, headphones on","unsupportedLabel":"3D rendering is unavailable on this device.","audioUnavailableHint":"Audio is unavailable on this device","previewUnavailableHint":"Preview unavailable · synthesised harmonic signature","synthHint":"Harmonic signature synthesised live"},"games":{"challengeTitlePrimary":"A challenge for everyone","challengeTitleSecondary":"your move","browseLabel":"Browse the games","duelTitlePrimary":"Challenge your friends","duelTitleSecondary":"and the whole world","leaderboardLabel":"View the leaderboard","gameLine":"Selected game: {name}, {kind}.","playAriaLabel":"{cta} — {name}","cta":{"takuzu":"Play Takuzu","hitori":"Play Hitori","nurikabe":"Play Nurikabe","sudoku":"Play Sudoku","gratte-ciel":"Play Gratte-ciel"},"kind":{"takuzu":"6 × 6 binary grid","hitori":"grid of digits","nurikabe":"islands ringed by the sea","sudoku":"9 × 9 grid","gratte-ciel":"towers and sightlines"},"challengeLead":"Five minutes are enough to improve, collect points and come back tomorrow.","duelLead":"Score points, keep your streak and share every win.","startLabel":"Start your progress","facts":{"newGame":"new game a month","levelsLead":"Thousands","levels":"of levels","hoursLead":"Hours","hours":"of logical fun","players":"players","friendsLead":"Challenges","friends":"with friends","streaksLead":"Streaks","streaks":"bonus"},"factsChallengeLabel":"What the Tangente games offer","factsCommunityLabel":"The Tangente Jeux community in figures","boardLabel":"World leaderboard for the week, ranks 4 to 8","boardTitle":"Leaderboard","boardPeriod":"this week","dotsLabel":"Choose a game","dotLabel":"Show {name}","stageGamesLabel":"Tangente games carousel. Left and right arrows to change game, Enter to open it.","stageBoardLabel":"Weekly podium. Left and right arrows to browse the top three, Enter to take the challenge.","unsupportedLabel":"3D rendering is unavailable on this device.","previousGameLabel":"Previous game","nextGameLabel":"Next game","podiumLine":"{rank} place — {name} · {points} points · {streak}-day streak.","podiumRank":{"1":"1st","2":"2nd","3":"3rd"}}}},"home":{"latestMathNews":"The latest in mathematics","continuous":" continuously","discover":"Discover ","gamesChallenges":"Games & Challenges","articlesDescription":"Dive into the fascinating world of mathematics with our captivating articles, written by passionate experts. Whether you are a curious amateur or a seasoned mathematician, our articles cover a wide range of topics, from fundamental concepts to the latest discoveries, including practical applications and philosophical reflections.","viewAllGamesChallenges":"View all Games & Challenges","noMagazinesFound":"No magazines found","noResultsFound":"No results found","magazine":"Magazine","tangenteMagazine":"Tangente Magazine","magazineDescription":"Discover how great mathematicians shaped our world, explore the links between math and other fields like art, music, or even philosophy, and relive the key moments that marked its evolution.","foldersTitle":"Folders","foldersDescription":"Discover how great mathematicians shaped our world, explore the links between math and other fields like art, music, or even philosophy, and relive the key moments that marked its evolution.","similarArticles":"Similar Articles","articleBy":"An article by","themes":"Themes","breadcrumbHome":"Home","breadcrumbArticles":"Articles","publishedIn":"Published in","browseIssue":"Browse the issue","folder":"Folder","folderArticleCount":"Series of {count} articles","articleCount":"{count, plural, one {# article} other {# articles}}","viewFullFolder":"View the full folder","issueAllFolders":"All folders","issueAllArticles":"All articles","ofThisIssue":"of this issue","readingNow":"Reading","article":"Article","author":"Author","sameThemeAccent":"On the same theme","sameThemeSubtitle":"Articles recommended for you.","listenEyebrow":"Listen to the article","listenPlay":"Play","listenPause":"Pause","listenPreview":"Preview","listenSubscribe":"Subscribe to listen","listenSpeed":"Speed","listenSeek":"Position in podcast","listenOnTangente":"on Tangente","listenOpenInAudio":"Open in Tangente Audio","listenNew":"New","mathThemesDescription":"Explore the major mathematical themes: geometry, algebra, analysis, arithmetic, logic, and many other fascinating fields.","subscribeCta":"Become a member to read this article and all of Tangente!","alreadySubscribedLogin":"Already subscribed?","loginLink":"Log in","noDigitalAccessTitle":"Content not included in your subscription","noDigitalAccessDescription":"Your current subscription does not include digital access to articles. Adding this option online is not yet available, but our support team can take care of it for you!","noDigitalAccessContact":"Call us at","noDigitalAccessOr":"or email us at","digitalAccessTroubleshootTitle":"Having trouble viewing this article?","digitalAccessTroubleshootDescription":"If the article isn't fully displayed, try signing out and signing back in.","digitalAccessTroubleshootPersist":"If the issue persists, email us at","digitalAccessTroubleshootOrReport":"or use the \"Report an issue\" button at the bottom of this page.","digitalAccessTroubleshootThanks":"Thank you for your understanding.","solutionDialogTitle":"Become a member","solutionDialogDescription":"Become a member to see the solution and all of Tangente","viewSolution":"View solution","solutionHeading":"Solution","llmAssistedNotice":"LLM-assisted article, produced under the author's scientific and editorial direction.","subscribeFeature1":"Unlimited access to all Tangente Magazine content","subscribeFeature2":"6 issues of Tangente in print or digital version","subscribeFeature3":"6 issues of Tangente Special in print or digital version","cancelAnytime":"Cancellable at any time.","subscribe":"I subscribe","renewSubscription":"Renew my subscription","renewSubscriptionShort":"Renew","pastDueTitle":"Your payment has failed","pastDueDescription":"Your last payment could not be processed. Renew your subscription to regain access to all articles.","category":"Category","all":"All","tangentNumbers":"Tangente numbers","specialIssues":"Special issues","sortBy":"Sort by","select":"Select","mostRecent":"Most recent","oldest":"Oldest","searchOnTangente":"Search on Tangente","gettingStarted":"Getting Started","guides":"Guides","apiReference":"API Reference","fullApiDocumentation":"Full API documentation","searchDocumentation":"Search documentation...","bannerTitle":"XXL Digital Subscription","bannerYears":"15 years of mathematical culture","bannerIssues":"38 issues + 24 Special Editions","bannerProblems":"420 world problems","bannerCta":"Access everything (€5/month)","latestIssues":"Our latest issues","promoTitle":"All of Tangente, everywhere","promoFeature1":"Unlimited access to all articles","promoFeature2":"New issues as soon as they're published","promoFeature3":"15 years of archives available","promoCta":"Discover our plans","latestIssueAriaLabel":"Issue {number}","byline":{"viewAuthorPage":"View {name}’s page","theAuthor":"the author","signedBy":"Written by","updatedLabel":"Updated","readMinutes":"{minutes} min read"},"share":"Share","copyLink":"Copy link","linkCopied":"Link copied","shareOnX":"X","shareOnFacebook":"Facebook","shareByEmail":"Email","glossaireTerms":"{count, plural, one {term} other {terms}}","glossaireEmpty":"No term found.","bannerLibraryTitle":"Bibliothèque Tangente","bannerLibraryFeature1":"The finest library of mathematical culture","bannerLibraryFeature2":"92 thematic volumes","bannerLibraryFeature3":"From artificial intelligence to fractals","bannerLibraryCta":"Subscribe","infinimath":{"intro":"Discover the","cta":"Visit the shop","comics":{"title":"Graphic Novels","subtitle":"Math comics for children & adults","cover1Alt":"Little Katherine Johnson","cover2Alt":"Perdus dans la 4e dimension","cover3Alt":"Attaque sur CyberSchool"},"games":{"title":"Games & Puzzles","subtitle":"Puzzles, challenges & logic","cover1Alt":"The Alchemist Set","cover2Alt":"The Alchemist Iron","cover3Alt":"Brain Puzzle"},"youth":{"title":"Children","subtitle":"Picture books & novels for children","cover1Alt":"La faiseuse de neige","cover2Alt":"Conte tout droit","cover3Alt":"TIROND"},"artBooks":{"title":"Illustrated Books","subtitle":"Gift ideas & culture","cover1Alt":"Voyages au pays des maths","cover2Alt":"Le Beau Livre des Maths","cover3Alt":"La belle histoire des maths"},"bridge":{"title":"Bridge","subtitle":"Books, magazines & software","cover1Alt":"Les maniements de couleur","cover2Alt":"La défense","cover3Alt":"Le jugement au bridge"}}},"header":{"subscribeCta":"Subscribe"},"issue":{"numberShort":"No. {number}"},"search":{"placeholder":"Search articles, folders, issues, authors...","noResults":"No results found","noResultsDescription":"Try different keywords","resultCount":"{count, plural, one {# result} other {# results}}","articles":"Articles","folders":"Folders","issues":"Issues","authors":"Authors","all":"All"},"footer":{"tagline":"A different perspective on mathematics.","home":"Tangente home","discoverSubscriptions":"Discover subscriptions","newsletter":{"title":"Stay curious.","description":"Fresh articles and magazine news, straight to your inbox.","emailLabel":"Email address","placeholder":"Your email address","submit":"Subscribe","note":"Free newsletter. Unsubscribe anytime.","success":"Thank you! You are subscribed to the Tangente newsletter."},"siteNavigation":"Site navigation","readTangente":"Read Tangente","latestArticles":"Latest articles","glossary":"Glossary","exploreMaths":"Explore mathematics","allThemes":"View all themes","forYou":"For you","faq":"Frequently asked questions","aboutTangente":"About Tangente","aboutMagazine":"About the magazine","copyright":"© {year} Tangente","legalNotice":"Legal notice","privacyPolicy":"Privacy policy","termsOfSale":"Terms and Conditions of Sale","cookies":"Cookies","language":"Language"},"layout":{"back":"Back","guestUser":"Guest User","notifications":"Notifications","myAccount":"My account","signIn":"Sign in","etablissementAccess":"Institution","account":{"manageSubscription":"Manage my subscription","logout":"Sign out","chipActive":"ACTIVE","chipEnding":"ENDING","activeUntil":"Active until {date}","endsOn":"Ends on {date}","renewNow":"Renew now","sepaManaged":"Collected automatically — nothing to do on your side.","settleInvoice":"Settle my invoice"}},"planChange":{"title":"Change Plan","description":"Compare plans and change your subscription.","noSubscription":"You don't have an active subscription.","discoverOffers":"Discover our offers","currentBadge":"Current","choosePlan":"Choose a plan","customizeChange":"Customize your change","duration":"Duration","options":"Options","shipping":"Shipping","shippingMetropole":"Mainland France","shippingMetropoleIncluded":"Included","shippingOverseas":"Overseas","shippingOverseasLineItem":"Overseas shipping","total":"Total","currentlyLabel":"Currently: {amount}{interval}","upgrade":"Switch to this plan","downgrade":"Downgrade","savingsText":"You save {amount} over {duration}","successToast":"Your subscription has been updated successfully.","errorToast":"Unable to change your subscription. Please try again.","loadingError":"An error occurred while loading plans. Please try again.","formulaPrefix":"Plan {name}","interval":{"monthly":"1 month","semester":"6 months","annual":"1 year","biennial":"2 years"},"intervalLabel":{"monthly":"/month","semester":"/semester","annual":"/year","biennial":"/2 years"},"shippingContactSupport":"Contact support to change your shipping zone.","upgradeImmediate":"Change takes effect immediately with prorated billing.","downgradeScheduled":"Change takes effect at the end of the current billing period.","downgradeSuccessToast":"Your downgrade has been scheduled for the end of the billing period.","previewButton":"Preview changes","confirmButton":"Confirm change","previewLoading":"Calculating...","previewChangeType":{"upgrade":"Upgrade","downgrade":"Downgrade","same":"No change"},"previewEffectiveDate":"Effective date","previewImmediate":"Immediate","previewBlocked":"This change cannot be applied","previewBackendUpgrade":"You will be charged the prorated difference immediately.","previewBackendDowngrade":"Your current plan remains active until the end of the billing period.","previewModify":"Modify selection","noChangeHint":"Select a different interval or toggle options to make a change."},"account":{"subscription":{"title":"Subscription","noSubscription":"No active subscription.","details":"Details of your current subscription.","plan":"Plan","currency":"Currency","periodStart":"Period start","periodEnd":"Period end","items":"Products","renewalTotal":"Renewal total","autoRenewal":"Automatic renewal at the end of each period.","autoRenewalSponsored":"Automatic renewal paid for by {payer}.","autoRenewalSponsoredAnonymous":"Automatic renewal paid for by the person giving it to you.","sponsored":{"badge":"Gift from {payer}","badgeAnonymous":"Gifted subscription","title":"Your subscription is a gift from {payer}","titleAnonymous":"Your subscription is a gift","renewing":"Renewed automatically by {payer}, until {date}.","renewingAnonymous":"Renewed automatically by the person giving it to you, until {date}.","endsOn":"Your gifted subscription ends on {date}. You can then choose your own plan.","paymentFailed":"{payer}'s payment failed. You can take the subscription over to keep your access.","paymentFailedAnonymous":"The payment of the person giving it to you failed. You can take the subscription over to keep your access.","takeoverCta":"Take the subscription over"},"sponsoredAddress":{"title":"Check your delivery address","description":"The person giving you the subscription entered this address. Check it in your addresses to receive your issues.","manage":"Manage my addresses"},"autoRenewalCard":"Automatic renewal at the end of each period, charged to your {brand} card ending {last4}.","autoRenewalCardGeneric":"Automatic renewal at the end of each period, charged to your card ending {last4}.","autoRenewalWallet":"Automatic renewal at the end of each period, through {wallet}.","autoRenewalSepa":"Automatic renewal at the end of each period, by SEPA direct debit ending {last4}.","autoRenewalPrelevement":"Automatic renewal at the end of each period, by SEPA direct debit from your bank account.","autoRenewalNone":"Your subscription does not renew automatically: no payment method on file.","autoRenewalNoneHint":"Add one in the “Payment method” section to enable automatic renewal.","autoRenewalExpired":"Your subscription does not renew automatically: your saved payment method has expired.","autoRenewalExpiredHint":"Update it in the “Payment method” section to resume automatic renewal.","manageSubscription":"Manage subscription","cancelButton":"Cancel my subscription","cancelDialogTitle":"Cancel your subscription?","cancelDialogDescription":"Your subscription will remain active until the end of the current period. You will no longer be charged after that date.","cancelReasonTitle":"Why do you want to cancel?","cancelReasonDescription":"Help us improve our service by telling us why you're canceling.","cancelReasons":{"too_expensive":"It's too expensive","not_using":"I'm not using the service enough","missing_content":"It's missing features I need","switched":"I'm switching to another provider","other":"Other reason"},"cancelReasonOtherLabel":"Specify your reason","cancelReasonOtherPlaceholder":"Tell us more...","cancelReasonNext":"Continue","cancelSubmit":"Cancel my subscription","cancelSupportInfo":"For more information, contact our customer service on","cancelKeepsAccessUntil":"Your subscription stays active until {date}. You will not be charged after that date.","cancelBackToSubscription":"Back to my subscription","cancelDoneTitle":"Your subscription has been cancelled","cancelDoneUntil":"You keep access until {date}. You will not be charged after that date.","cancelTrialEndsNow":"Your trial ends immediately: access and the remaining issues are withdrawn as soon as you confirm. This cannot be undone.","cancelTrialDone":"Your trial has ended. You will not be charged.","cancelDoneUndoHint":"You can reverse this from your subscription page for as long as the current period is running.","cancelError":"Your cancellation could not be recorded. Try again or contact our customer service.","cancelDialogConfirm":"Confirm cancellation","cancelDialogCancel":"Keep my subscription","cancelDialogBack":"Back","planChangeScheduled":"Plan change scheduled: your new plan starts on {date}. Your current plan stays active until then.","planChangeScheduledTitle":"A plan change is scheduled","planChangeScheduledLink":"View the scheduled change","planChangeScheduledUndated":"Plan change scheduled: your new plan starts at the end of your current period.","cancellationScheduled":"Cancellation scheduled on {date}","cancelPasswordTitle":"Security","cancelPasswordDescription":"To confirm it's really you ending this subscription, please enter your password.","cancelPasswordLabel":"Password","cancelPasswordSubmit":"Confirm your password","cancelPasswordIncorrect":"Incorrect password. Please try again.","cancelPasswordError":"We could not verify your password. Try again or contact our customer service.","cancelConfirmationExpired":"Your confirmation has expired or was already used. Please enter your password again to complete the cancellation.","cancelBack":"Back","cancelConfirmTitle":"Confirm the end of your subscription","cancelConfirmDescription":"Confirming ends your subscription: it will not be renewed.","cancelReasonSkip":"Skip this step","undoCancellation":"Undo cancellation request","quotaTitle":"Quota usage","quotaDescription":"Number of paper magazines sent compared to the total included in your subscription.","quotaEmpty":"No quota available.","changeInfoOr":"or by email at","changePlanTitle":"Upgrade your subscription","changePlanDescription":"Compare the available plans and pick the one that best fits how you read.","changePlanChangeTermCta":"Change billing term","changePlanCta":"Choose this plan","changePlanCurrentBadge":"Your current plan","changePlanFromPrice":"From","changePlanDiscount":"-{percent}%","changePlanLoadError":"Plans could not be loaded. Try again in a moment or contact our customer service team.","changePlanEmpty":"No plan is available right now.","changePlanEffectiveOn":"The change takes effect on {date}, at the end of your current period.","changePlanUnavailable":"Changing plan online is not open yet. Our customer service team can do it for you.","changePlanRequested":"Your plan change is recorded.","changePlanCancelled":"The plan change has been cancelled.","changePlanAddPaymentMethod":"Add a payment method","changePlanPendingBadge":"Scheduled","changePlanPendingTitle":"Plan change scheduled","changePlanPendingDescription":"Your subscription switches to your new plan on {date}. Nothing changes until then.","changePlanPendingDescriptionUndated":"Your subscription switches to your new plan at the end of your current period.","changePlanPendingCancel":"Cancel this change","changePlanBlocked":{"no_subscription":"You have no active subscription.","ambiguous_subscription":"Several subscriptions are attached to your account: our customer service team can make the change for you.","not_active":"Your subscription is not active.","not_digital":"Changing plan online only applies to digital subscriptions.","unknown_plan":"Your current plan is not recognised: our customer service team can make the change for you.","canceled":"Your subscription is already cancelled: it ends at the current expiry date.","change_pending":"A plan change is already scheduled on your subscription.","renewal_prepaid":"Your next term is already paid. Changing plan will be possible from the following period.","renewal_invoice_open":"A renewal invoice is awaiting payment. Settle it before changing plan.","no_period_end":"Your subscription expiry date is unavailable: our customer service team can make the change for you.","prelevement":"Your subscription is paid by direct debit: our customer service team can make the change for you.","no_payment_method":"No usable card is saved on your account. Add a payment method to change plan.","etablissement_seat":"Your access is provided by your institution: plan changes go through them.","switch_too_far_ahead":"Your expiry date is still too far away. Changing plan will be possible closer to the end of your period.","switch_too_soon":"Your expiry date is too close to schedule the change. Our customer service team can help.","no_targets":"No other plan is available for your subscription at the moment.","generic":"Changing plan is not available for your subscription. Our customer service team can help."},"changePlanError":{"notEligible":"The change is not possible on your subscription anymore.","noPendingChange":"No plan change is scheduled.","invalidTarget":"This plan is not available.","notCancellable":"This change can't be cancelled anymore.","deliveryAddressRequired":"A delivery address is required for a print plan. Please contact our customer service team.","rateLimited":"Too many attempts. Try again in a few minutes.","generic":"The operation failed. Try again in a moment."},"paymentProcessing":"Your payment is being processed. Your access will be activated as soon as the payment is confirmed.","paymentProcessingSepa":"Your SEPA direct debit payment is being processed. Your access will be activated as soon as the payment is confirmed, within a few business days.","renewal":{"renewTitle":"Renew your subscription","renewDescription":"Pay for your next period now to keep your access. You will be redirected to a secure payment page.","renewCta":"Renew now","renewCtaWithAmount":"Renew now ({amount})","pendingTitle":"An invoice is awaiting payment","pendingDescription":"Pay your invoice to complete your subscription renewal.","payPending":"Pay my invoice","recoverTitle":"Settle your payment","recoverDescription":"Your last invoice payment failed. Pay it to keep your access.","recoverCta":"Pay the invoice","resubscribeTitle":"Resume your subscription","resubscribeDescription":"Your subscription has ended. Pick up right where you left off in just a few clicks.","resubscribeCta":"Resume my subscription","paymentLinkError":"Could not retrieve the payment link. Please try again.","sepaManagedTitle":"Your subscription is paid by automatic direct debit","sepaManagedDescription":"Your subscription is paid by SEPA direct debit from your bank account. There is nothing for you to do. For any questions or to update your bank details, contact our customer support.","sepaRejectedDescription":"Your last SEPA direct debit did not go through. Contact our customer support to update your bank details, or pay your invoice by card now.","sepaRejectedNoInvoiceDescription":"Your last SEPA direct debit did not go through. Contact our customer support to update your bank details and settle your account.","sepaManagedCancelingDescription":"Your subscription is paid by SEPA direct debit. Its cancellation is scheduled: it will end on the date shown above.","sepaRejectedCta":"Pay by card","uncancelTitle":"Resume your subscription","uncancelDescription":"Your subscription will end on {date}. Undo the cancellation to continue without interruption — it's free, no payment required.","uncancelDescriptionNoDate":"Your subscription is canceled at the end of the current period. Undo the cancellation to continue without interruption — it's free, no payment required.","uncancelCta":"Resume my subscription","uncancelError":"Could not undo the cancellation. Please try again."},"paymentMethodTitle":"Payment method","paymentMethodDescription":"View or update your bank card or SEPA direct debit.","paymentMethodUpdate":"Update payment method","paymentMethodCard":"{brand} card •••• {last4}","paymentMethodWallet":"Paid through {wallet}","paymentMethodSepa":"SEPA direct debit •••• {last4}","digitalAccessLabel":"Digital access"},"customerNumber":"Your subscriber number","preferredLanguage":"Preferred language","profileUpdated":"Profile updated","deleted":"Account deleted","access":{"title":"Your digital access","grantedByEtablissement":"Access provided by {name}.","etablissementFallback":"your institution","grantedByGeneric":"Your digital access is active.","until":"Until {date}.","noPersonalSubscription":"You have no personal subscription: this access is not billed to you and there is nothing to cancel."},"addresses":{"deliveryCountryLocked":"To change the delivery country, contact support."},"gifts":{"title":"My gifts","empty":"No gifts yet.","paidOn":"Ordered on {date}","statusPending":"Being prepared","statusFulfilled":"Sent","issueNumber":"No. {number}","quantity":"× {quantity}"},"sponsorships":{"title":"Gifted subscriptions","intro":"The subscriptions you give. You can stop their renewal at any time.","empty":"You have not given any subscription yet.","offerCta":"Give a subscription","loadError":"We could not load your gifted subscriptions.","offeredOn":"Given on {date}","planLabel":"Plan:","amountLabel":"Renewal amount:","nextRenewal":"Next renewal on {date}","activeUntil":"Active until {date}","status":{"active":"Active","pending_payment":"Payment pending","ended":"Ended","abandoned":"Payment not completed","stopped":"Renewal stopped"},"pendingNote":"We are waiting for your payment to be confirmed.","notClaimed":"{email} has not activated their account yet.","anonymousNote":"Your name is not revealed to the recipient.","resumeRenewal":"Resume the renewal","changeCard":"Change card","manage":"Manage","manageTitle":"Manage the gifted subscription","manageDescription":"Given to {email}","cardLabel":"Card:","cardExpires":"expires end of {date}","paymentOutstandingNotice":"The last renewal could not be paid. Change the card to pay it and keep the subscription active.","backToList":"Back to gifted subscriptions","backToManage":"Back to the gifted subscription","cancel":"Cancel the subscription","cancelKeep":"Keep the gifted subscription","cancelConfirmTitle":"Confirm the cancellation","cancelConfirmDescription":"By confirming, the subscription given to {email} will not renew.","cancelKeepsAccessUntil":"It stays active until {date}. You will not be charged after that date.","cancelSubmit":"Cancel the gifted subscription","cancelDoneTitle":"The gifted subscription is cancelled","cancelDoneUntil":"The subscription given to {email} stays active until {date}. You can reactivate it until then.","cancelDone":"The subscription given to {email} will not renew. You can reactivate it while it is active.","notCancellable":"This gifted subscription cannot be cancelled online for now.","notFound":"This gifted subscription cannot be found.","cardUpdated":"Your new card is being saved. It will be used for upcoming payments.","resume":{"title":"Resume the renewal?","description":"The subscription given to {email} will renew automatically again on your card.","confirm":"Resume"},"card":{"title":"Change card","description":"You will be taken to our secure payment page to save the card that pays for the subscription given to {email}. Any outstanding payment will be settled with it.","confirm":"Continue"},"passwordLabel":"Password","passwordHint":"For your security, confirm with your account password.","errors":{"passwordIncorrect":"Incorrect password. Please try again.","confirmationExpired":"Your confirmation has expired. Enter your password again.","paymentOutstanding":"The last renewal is unpaid. Change the card to settle it before stopping the renewal.","notAllowed":"This action is not possible for this subscription. Contact us if needed.","takeoverUnavailable":"This subscription cannot be taken over online. Contact us and we will sort it out with you.","rateLimited":"Too many attempts. Try again in a few minutes.","redirectFailed":"The redirect to the payment page failed. Please try again.","stoppedByBeneficiary":"The recipient stopped the renewal: it cannot be resumed.","generic":"Something went wrong. Please try again."},"stoppedByBeneficiary":"The recipient stopped the renewal.","beneficiary":{"takeover":{"title":"Take the subscription over?","description":"The gifted subscription stops and you are taken to our secure payment page to subscribe in your own name.","confirm":"Continue"}}},"sessionUnavailable":"Connection temporarily unavailable.","nav":{"profile":"Profile","security":"Security","addresses":"Addresses"}},"pricing":{"alreadySubscribedTitle":"You are already subscribed","alreadySubscribedDescription":"You already have an active subscription. Visit your subscription settings to manage your plan.","subscribedWithoutDigitalContact":"Call us at","subscribedWithoutDigitalOr":"or email us at","goToSubscription":"Go to my subscription","paymentIssueTitle":"Your payment did not go through","paymentIssueDescription":"Your subscription is still running, but a payment was not settled. Pay the outstanding invoice to restore your access — subscribing again would bill you for two subscriptions.","statusUnknownTitle":"We could not check your subscription","statusUnknownDescription":"We cannot read your subscription status right now. Try again in a moment: we would rather wait than risk billing you for a second subscription.","statusUnknownRetry":"Try again","recoveryRequiredNotice":"Your account already has a subscription awaiting payment. Settle the outstanding invoice to reactivate it — subscribing again would bill you for two subscriptions.","sepaManagedNotice":"Your current subscription is paid by SEPA direct debit from your bank account: there is nothing for you to do to renew it. Contact customer service with any questions.","auth":{"createAccount":"Create my account","createAccountAndPay":"Create my account and pay","createAccountAndGift":"Create my account and give","creatingAccount":"Creating your account…","passwordPlaceholder":"At least 8 characters","errors":{"credentialsRequired":"Email and password are required","invalidCredentials":"Incorrect email or password","loginFailed":"Sign-in error","loginUnavailable":"Sign-in service unavailable","emailRequired":"Please enter an email address","passwordTooShort":"The password must be at least 8 characters long","emailAlreadyUsed":"An account already exists with this email address. Please sign in.","registrationFailed":"Could not create the account","autoLoginFailed":"Account created, but automatic sign-in failed. Please sign in manually.","serviceUnavailable":"Service unavailable","redirectFailed":"The redirect to payment failed. Please try again.","sessionExpired":"Your session has expired. Please sign in again."},"registerTab":"Create your account","loginTab":"You already have an account","emailLabel":"Email address *","emailPlaceholder":"your@email.com","emailHint":"Check your email address carefully — it is what you will use to sign in to your account.","passwordLabel":"Password *","loginPasswordPlaceholder":"Your password","forgotPassword":"Forgot your password? Click here to reset it.","signIn":"Sign in","google":"Continue with Google","or":"or","emailLockedHint":"This is your order’s address: the account must carry it to activate the offer.","existingAccountHint":"A magazine account may already exist for this address: the link above sets a password if none was ever set."},"gift":{"badge":"Free gift","offerClaim":"Included with every new subscription","metropoleOnly":"Not available outside mainland France.","description":"Choose your welcome gift.","summaryNote":"Your gift will be shipped as soon as your subscription is confirmed.","loginToChoose":"Sign in to choose your issues","condition":{"sentence":"For every new {hasIntervals, select, yes {{intervals} } other {}}subscription{hasPlanTypes, select, yes { on the {planTypes} plan} other {}}.","interval":"{interval, select, monthly {monthly} semester {6-month} annual {1-year} biennial {2-year} other {}}","planType":"{planType, select, paper {paper} digital {100% digital} other {}}","separator":", "},"errors":{"required":"Choose at least one issue for your gift.","droppedContinuing":"We could not save this gift. We are continuing with your subscription without it.","declineFailedContinuing":"We could not cancel your gift. Your subscription continues, and the gift already saved may still be sent.","savedGiftLocked":"Your gift has already been saved for this subscription and can no longer be cancelled from this page. Choose at least one cover to continue, or contact us."},"payWithoutGiftCta":"Pay without the gift","status":{"paidTitle":"Your gift is confirmed","paidDescription":"Your gift will be shipped shortly. There's nothing else for you to do.","paidContactDescription":"Your gift will be shipped shortly. Our team will contact you to finalize delivery.","pendingTitle":"Gift being confirmed","pendingDescription":"We're finalizing your gift. There's nothing else for you to do.","cancelled":{"nothingOwedTitle":"Gift not awarded","nothingOwedDescription":"This gift came with a payment. Nothing was due for this operation, so the gift was not awarded. Your subscription is up to date.","paidElsewhereTitle":"Gift not awarded","paidElsewhereDescription":"Your payment was received, but it was settled outside the online journey, so this gift cannot be awarded automatically. Get in touch if you believe you are entitled to it.","offerWithdrawnTitle":"Gift offer withdrawn","offerWithdrawnDescription":"This gift offer is no longer available. Your subscription is unaffected.","paymentIncompleteTitle":"Gift not awarded","paymentIncompleteDescription":"The associated payment did not go through, so the gift was not awarded.","giftUnavailableTitle":"Gift not awarded","giftUnavailableDescription":"We could not award this gift. Your subscription is recorded and up to date."}}},"paywall":{"titleLead":"Choose your","titleAccent":"Tangente subscription","subtitle":"Maths to read, listen to and play.","loading":"Loading plans…","loadError":"Something went wrong while loading the plans. Please try again.","tabsLabel":"Subscription type","tabs":{"digital":"100% digital","paper":"Print + digital"},"included":{"digitalEyebrow":"Included in the digital subscription","paperEyebrow":"Digital access included in every print subscription","tangenteAlt":"Tangente","jeuxAlt":"Tangente Jeux"},"intervalGroupLabel":"Subscription length","bestOffer":"Popular","introFirstMonth":"{amount} for the first month","introThen":"Then {amount} / month — no commitment","perMonth":"that's {amount} / month","continueCta":"Continue my order","reassurance":{"securePayment":"Secure payment","noCommitment":"No commitment","cancelAnytime":"Cancel any time","freeDelivery":"Delivery included in France","autoRenew":"Automatically renewed"},"paper":{"fromPrice":"from","chooseFormula":"Choose this plan","selected":"Selected","digitalIncluded":"Digital included","back":"Back","durationTitle":"Choose how long you subscribe for","durationReminder":"{plan} plan","discountPercent":"-{percent}%","discountSavings":"saved","perUnit":"{interval, select, monthly {/ month} semester {/ 6 months} annual {/ year} biennial {/ 2 years} other {}}","chooseFormulaFor":"Choose the {plan} plan","frenchOnlyNotice":"Tangente Paper is published in French only.","frenchOnlyTitle":"Choose your subscription"},"duration":{"term":"{interval, select, monthly {1 month} semester {6 months} annual {1 year} biennial {2 years} other {}}"},"faq":{"title":"Your questions, answered","lead":"The questions we are asked most often about Tangente subscriptions.","help":"Another question? Email us at {emailAddress} or call {phoneNumber}."},"finalCta":{"title":"Ready to subscribe?","lead":"Start at €1 for the first month. No commitment. Read straight away.","cta":"See the plans"}},"recap":{"eyebrow":"Your order summary","modify":"Edit","signedIn":"You are signed in","switchAccount":"Switch account","durationLabel":"Subscription length","autoRenewNote":"(renews automatically)","deliveryLabel":"Delivery","deliveryMetropolitan":"Mainland France","deliveryOverseas":"Outside mainland France","giftIncluded":"Gift included","giftShippedImmediately":"Shipped immediately","giftMoreCovers":"+{count}","basePrice":"Base subscription","overseasShippingSurcharge":"Delivery surcharge outside mainland France","firstMonthDiscount":"First-month discount","firstMonthTotal":"First month total","totalTtc":"Total incl. VAT","freeTrial":"Free trial","free":"Free","recurringAmount":"{amount} / {interval, select, monthly {month} semester {6 months} annual {year} biennial {2 years} other {}}","thenRecurringAmount":"then {amount} / {interval, select, monthly {month} semester {6 months} annual {year} biennial {2 years} other {}}","addPromoCode":"Add a promo code","promoPlaceholder":"Promo code","promoApply":"Apply","promoInvalid":"Invalid promo code","promoNotApplicable":"This code does not apply to the selected offer.","trialDays":"{count, plural, one {# free day} other {# free days}}","trialMonths":"{count, plural, one {# free month} other {# free months}}","payNow":"Pay now","giftNow":"Give Tangente","payRedirecting":"Redirecting…","startTrial":"Start my trial","loginToPay":"Sign in to pay","genericError":"Something went wrong. Please try again.","statusReadFailed":"We cannot check your subscription right now. Try again in a moment before paying.","statusChecking":"Checking your subscription…","securePaymentEyebrow":"Secure payment","securePayment":"100% secure payment.","legalSentence":"By confirming your subscription, you acknowledge that you have read the terms of sale and the privacy policy, and that you accept them.","cardLabel":"Bank card","helpLine":"Need help? Call us on","durationValue":"{interval, select, monthly {1 month} semester {6 months} annual {1 year} biennial {2 years} other {}}","pageTitle":"Your details and payment","pageSubtitle":"Complete your subscription","deliveryAddress":"Delivery address","billingAddress":"Billing address","differentBillingAddress":"Different billing address"},"sponsor":{"offerTitle":"Why not give this subscription as a gift?","subtitle":"Treat someone close to you to Tangente.","offerCta":"Give this subscription","subscriberOfferCta":"Give a subscription","activeTitle":"You are giving this subscription","forMe":"It's for me","emailLabel":"Recipient's email address","emailPlaceholder":"firstname@example.com","emailHint":"We will use this email to create their account, give them digital access and tell them about your gift.","emailRequired":"The recipient's email address is required","emailInvalid":"Invalid email address","deliveryTitle":"Who are you giving Tangente to?","deliverySubtitle":"The magazines will be sent to this person.","firstNameLabel":"Recipient's first name","lastNameLabel":"Recipient's last name","addressLabel":"Recipient's address","messageLabel":"A short note for the recipient (optional)","billingInvoiceNote":"You enter your billing details on the secure payment page and receive the payment receipt in your name. The subscription's invoice is issued to the recipient.","errors":{"refused":"We cannot give this subscription to that address. Contact us if needed.","selfGift":"You cannot give a subscription to yourself.","notSelectable":"This plan cannot be given for this delivery address. Please check the recipient's country.","invalid":"Check the recipient's details, then try again.","rateLimited":"Too many attempts. Try again in a few minutes.","unavailable":"Giving a subscription is not available at the moment."}},"address":{"firstNameRequired":"First name is required","lastNameRequired":"Last name is required","line1Required":"Address is required","cityRequired":"City is required","postalCodeRequired":"Postal code is required","countryRequired":"Country is required","firstNamePlaceholder":"John","lastNamePlaceholder":"Smith","line1Placeholder":"123 Main Street","line2Label":"Address line 2 (optional)","line2Placeholder":"Apartment, building, etc.","postalCodePlaceholder":"75001","cityPlaceholder":"Paris","countrySearchPlaceholder":"Search for a country..."},"fromPrice":"From","legacy":{"selectPlanFirst":"Please choose a plan first","title":"Choose your Tangente subscription","subtitle":"Join 6,000+ maths enthusiasts","formatQuestion":"Print or digital subscription?","reassurance":"No commitment · Cancel at any time","digitalBadge":"Digital","digitalIncluded":"Digital access included","fromPrice":"From","perSemester":"/ semester","thenPerInterval":"Then {amount}/{interval}, no commitment","thenPerMonth":"Then {amount}/month, no commitment","firstMonthThen":"for the first month, then {amount}/month, no commitment","selectPlan":"Please select a plan"},"paperFrenchOnly":{"question":"Is the print magazine available in English?","answer":"No. Tangente is published in French. A print subscription is delivered in French, and the digital access included with it is French as well."}},"checkout":{"successTitle":"Thank you for subscribing!","successDescription":"Your payment has been processed. You can now enjoy all your content.","goToSubscription":"Go to my subscription","backToHome":"Back to home","giftSuccessTitle":"Thank you! Your gift is on its way.","giftSuccessDescription":"We will let the recipient know by email as soon as the payment is confirmed.","giftGoToSponsorships":"Track my gifted subscriptions","cancelTitle":"Payment cancelled","cancelDescription":"Your payment was not completed. You have not been charged. You can go back to our plans and try again at any time.","cancelRetryCta":"Try the payment again"},"banner":{"deliveryAddress":{"message":"Your subscription includes paper magazines. Please provide your delivery address to receive them.","action":"Update my address"},"verifyEmail":{"message":"Your email address hasn't been confirmed yet. Check your inbox (and spam folder) for the confirmation link.","action":"Resend the email","sent":"Sent! The link is waiting in your inbox.","throttled":"You've reached the resend limit. Please wait a few minutes and try again.","error":"Sending failed. Please try again.","dismiss":"Dismiss"}},"support":{"reportIssue":"Report an issue","reportDescription":"Found an error in this article? Help us improve it.","issueType":"Issue type","category":{"typo":"Typo","incorrectContent":"Incorrect content","missingContent":"Missing content","mathFormula":"Math formula","other":"Other"},"describeIssue":"Describe the issue","descriptionPlaceholder":"Describe the issue you found (location, details...)","descriptionMinLength":"Description must be at least 10 characters.","send":"Send","sending":"Sending...","reportSuccess":"Your report has been sent.","reportError":"An error occurred. Please try again.","thankYou":"Thank you!","reportSuccessDescription":"We received your report and will review it shortly.","close":"Close","authRequired":"You must be logged in to report an issue.","loginOrCreate":"Log in or create an account"},"cookies":{"title":"We use cookies","description":"This site uses cookies to measure audience, improve your experience, and (with your consent) personalize ads.","accept":"Accept all","reject":"Reject all","personalize":"Personalize","save_preferences":"Save preferences","manage":"Manage cookies","necessary_label":"Strictly necessary","necessary_description":"Required for the site to function (session, security). Always enabled.","analytics_label":"Analytics","analytics_description":"Aggregated usage data to help us improve the site (PostHog, Google Analytics).","marketing_label":"Advertising","marketing_description":"Personalized ads and conversion measurement (Google Ads, AdSense, Meta Pixel)."},"newsletter":{"title":"Newsletter","search":{"placeholder":"Search by email"},"column":{"email":"Email","status":"Status","source":"Source","locale":"Language","createdAt":"Subscribed on"},"filter":{"source":"Source"},"source":{"salon2026":"Salon 2026","publicPopup":"Website popup","homepageFooter":"Homepage footer"},"action":{"enable":"Activate","disable":"Deactivate","delete":"Delete"},"status":{"active":"Active","inactive":"Inactive"},"confirm":{"deleteTitle":"Delete the subscription for {email}?","deleteDescription":"This action cannot be undone. The subscription will be removed from the list."},"toast":{"deleted":"Subscription deleted.","deleteFailed":"Deletion failed.","enabled":"Subscription activated.","enableFailed":"Activation failed.","disabled":"Subscription deactivated.","disableFailed":"Deactivation failed."},"popup":{"eyebrow":"BEFORE YOU GO","titlePrefix":"The Tangente Newsletter,","titleAccent":"every month.","lede":"Get the best of Tangente straight to your inbox: new puzzles, early-access content and a look behind the scenes at the editorial team.","benefit1":{"title":"3 brand-new puzzles","sub":"To rack your brain every month."},"benefit2":{"title":"An early-access article","sub":"Before everyone else, newsletter-exclusive."},"benefit3":{"title":"Behind the scenes","sub":"What's happening in the editorial team."},"input":{"placeholder":"Your email address"},"cta":"Subscribe","reassure":"No spam. Unsubscribe in one click.","success":{"title":"Thank you!","body":"You're now subscribed to the Tangente newsletter with {email}. See you soon in your inbox!"},"close":"Close","closeCountdown":"Closing available soon","error":{"generic":"An error occurred. Please try again in a moment.","rateLimited":"Too many attempts, please try again later."}}},"portal":{"userMenu":{"signOut":"Sign out"},"welcome":{"title":"Welcome to the portal","description":"The new admin portal is under construction. Use the sidebar to navigate the available sections."},"error":{"title":"Something went wrong","description":"This page could not be displayed. You can try again.","retry":"Try again"},"brand":{"name":"POLE Editions","workspace":"Workspace"},"reviewQueueBadge":{"ariaLabel":"{count, plural, one {# translation to review} other {# translations to review}}"},"header":{"unsavedChanges":"Unsaved changes","moreActions":"More actions","discardTitle":"Leave without saving?","discardDescription":"Your changes have not been saved and will be lost if you leave this page.","discardConfirm":"Leave without saving"},"form":{"settings":"Settings"},"translations":{"cardTitle":"Translations","pending":"Tracked in the review queue","openReviewQueue":"Open the review queue","source":"Source","switchToLocale":"Edit in {locale}"}},"translationStatus":{"not_translated":"Not translated","draft":"Draft","reviewed":"Reviewed","published":"Published","stale":"Stale"},"translationProvenance":{"ai":"AI","human":"Manual"},"problemes":{"breadcrumbHome":"Home","breadcrumbProblems":"Problems","reader":{"back":"Back to home","home":"Home","search":"Search","account":"My account","subscribe":"Subscribe","rights":"All rights reserved","notFoundTitle":"Problem not found","notFoundBody":"This problem does not exist or is no longer available."},"enonce":{"title":"Statement","empty":"This statement is empty."},"questions":{"title":"Questions"},"question":{"passed":"Question solved","locked":"Solve the previous questions first to unlock this one.","tag":"Q{n}","answer":"Answer","getHints":"Get hints"},"answer":{"label":"Your answer","placeholder":"Write your answer…","submit":"Check","submitting":"Grading…","empty":"Enter an answer before checking.","regrade":"Retry grading","verifyQuestion":"Check question {n}","analyzing":"Analysing…","collapse":"Collapse","lockedAfterSolution":"Answer locked — solution viewed. The assistant is still available to explain it."},"verdict":{"vrai":"Correct","vrai_incomplet":"Correct but incomplete","faux":"Incorrect"},"grading":{"pending":"Grading…","failed":"Grading is momentarily unavailable. Try again in a moment."},"hints":{"title":"Hints","reveal":"Show a hint","revealing":"Loading…","exhausted":"All hints are shown.","rank":"Hint {rank}","none":"No hints for this question.","restored":"Unlocked in a previous session.","intro":"Hints from the lightest to the closest to the solution. Reveal them sparingly — each revealed hint is final.","counter":"{opened}/{total} revealed","nextPrompt":"Click to reveal this hint.","revealThis":"Reveal this hint","lockedItem":"Locked — reveal the previous ones first.","askAi":"Ask the Tangente AI for help","askAiPrompt":"“{hint}”: can you expand on this hint?"},"chat":{"open":"Ask the assistant","title":"Assistant","close":"Close","loading":"Loading the conversation…","empty":"Start the conversation with the assistant.","placeholder":"Ask your question… (Enter to send)","send":"Send","sending":"Sending…","closed":"The conversation is over.","roleUser":"You","roleAssistant":"Assistant","remaining":"{count, plural, =0 {No messages left} one {# message left} other {# messages left}}","replyFailed":"Reply unavailable","intro":"The AI guides you with questions and leads — never the solution.","emptyTitle":"Ask your question.","emptyExample":"For example: “Where do I start?” or “Can you give me a lead without the answer?”"},"solution":{"reveal":"View the solution","revealing":"Loading…","title":"Solution","detailedTitle":"Detailed solution","empty":"No solution available.","confirmEyebrow":"Giving up already?","confirmTitle":"You're about to reveal the solution.","confirmBody":"Take a few more minutes — a hint or a nudge from the AI is often all it takes to get unstuck.","confirmKeepSearching":"Keep searching","confirmProceed":"Yes, show the solution","paywallEyebrow":"Subscribers only","paywallTitle":"This solution is in the issue.","paywallBody":"Subscribe to Tangente to read the full worked solution to this problem — and every other one.","paywallDismiss":"Later","summaryEyebrow":"Solution summary","fullEyebrow":"Full solution","showFull":"See the full demonstration","back":"Back to my notes"},"authGate":{"title":"Log in to solve","description":"Create an account or log in to use the hints, the assistant, and check your answers.","subscribe":"Subscribe","login":"Log in"},"error":{"authRequired":"Log in to continue.","rateLimited":"Too many attempts. Wait a moment before trying again.","tutorUnavailable":"The assistant is momentarily unavailable. Try again in a moment.","solutionAlreadyViewed":"You've viewed the solution — answering is locked. The assistant is still available to explain it.","solutionPrerequisites":"Resolve the previous questions first to access this solution.","generic":"Something went wrong. Please try again.","solutionLocked":"Solve the question or subscribe to access the solution.","questionLocked":"Solve the previous questions before this one.","allHintsUnlocked":"All hints are already shown.","interactionDisabled":"Interactions are not available for this problem.","answerTooLong":"Your answer is too long. Shorten it.","notRegradable":"This answer cannot be re-graded.","chatCapReached":"You have reached the message limit with the assistant.","messageTooLong":"Your message is too long. Shorten it."},"level":{"all_public":"All public","lycee":"High school","superieur":"Higher education","expert":"Expert"},"theme":{"logic":"Logic","arithmetic":"Arithmetic","algebra":"Algebra","geometry":"Geometry","numbers":"Numbers","combinatorics":"Combinatorics","probability":"Probability","analysis":"Analysis","game_theory":"Game theory"},"kind":{"logique":"Logic","curiosites_numeriques":"Numerical curiosities","geometrie":"Geometry","constructions_et_pavages":"Constructions and tilings","strategie_et_jeux":"Strategy and games","denombrement":"Counting","mise_en_equation":"Equation setup"},"header":{"eyebrow":"Problem:","share":"Send this problem to a friend","shareCopied":"Link copied","print":"Print"},"verdictFeedback":{"loading":"The Tangente AI is analysing your answer…","coachLabel":"Need a nudge?","askHelp":"Ask the AI for help","askHint":"Get a hint"},"atelier":{"title":"Solving help","subtitle":"Your notes stay on your device.","recallLabel":"Question recall","tabChat":"AI assistant","tabIndices":"Hints","close":"Collapse solving help","fab":"Get help"}},"seoPanel":{"title":"SEO","description":"Metadata for the public page. Leave empty to follow the source.","descriptionOwnCopy":"Metadata for the public page. Leave empty to use the page's own name and tagline.","metaTitle":"SEO title","metaTitlePlaceholder":"Title shown in search results","metaDescription":"SEO description","metaDescriptionPlaceholder":"Summary shown in search results","keywords":"SEO keywords","keywordsPlaceholder":"Add a keyword...","keywordsHint":"Never shown to visitors — distinct from the theme chips on the page.","inheritedHint":"Inherited from the source. Leave empty to keep following it.","charactersUsed":"{used}/{max} characters"},"games":{"entity":"game","list":{"title":"Games catalogue","search":"Search for a game..."},"details":{"title":"Editorial content"},"fields":{"name":"Name","gameKey":"Game key","tagline":"Tagline","taglineHint":"Short line shown on the game card.","description":"Description","quickRules":"Quick rules","quickRulesHint":"One rule per line. Leave empty to hide the panel.","badge":"Badge","enabled":"Enabled","displayOrder":"Display order","visibleFrom":"Visible from"},"status":{"enabled":"Enabled","disabled":"Disabled"},"state":{"title":"Catalogue state","description":"Commercial availability — distinct from the editorial content.","enabledHint":"When disabled the game leaves the public catalogue and becomes unplayable.","displayOrderHint":"Lowest first. The scale leaves gaps so a game can be inserted without renumbering.","visibleFromHint":"Paris time. The game appears from this date onwards.","missingPermission":"Administrators only: you can edit the content, not whether it is live."},"cover":{"title":"Cover","description":"Square artwork shown in the catalogue."},"ogImage":{"title":"Share image","description":"Shown when the page is shared on social networks (1200×630)."},"image":{"uploadError":"The image upload failed."},"error":{"notFound":"This game no longer exists.","forbidden":"You do not have permission to make this change."}},"renew":{"payableTitle":"Renew your subscription","payableDescription":"Check the summary below, then pay for your next period. You will be redirected to a secure payment page.","payCta":"Continue to payment","factPlan":"Plan","factInterval":"Billing period","factAmount":"Amount","factRenewalDate":"Due date","factEmail":"Account","intervalYear":"Yearly","intervalMonth":"Monthly","intervalOther":"As per your plan","reactivateTitle":"Reactivate your subscription","reactivateDescription":"Your subscription ends on {date}. Cancel the scheduled termination to continue without interruption — it is free, no payment required.","reactivateDescriptionNoDate":"Your subscription is set to end at the close of the current period. Cancel the scheduled termination to continue without interruption — it is free, no payment required.","reactivateCta":"Reactivate my subscription","reactivatedTitle":"Your subscription is reactivated","reactivatedDescription":"The cancellation has been reversed. Your subscription continues and will renew on {date}.","reactivatedDescriptionNoDate":"The cancellation has been reversed. Your subscription continues as normal.","alreadyRenewedTitle":"Your subscription is already renewed","alreadyRenewedDescription":"Nothing to do: your access is secured until {date}.","alreadyRenewedDescriptionNoDate":"Nothing to do: your subscription is up to date.","supersededTitle":"This link no longer matches your current subscription","supersededDescription":"Your subscription has changed since this email was sent. You will find its current state in your account.","resubscribedTitle":"Thank you, your subscription is in place","resubscribedDescription":"Your payment is confirmed and your subscription starts now. A summary email is on its way.","loginToSeeCta":"Sign in to see my subscription","confirmingTitle":"Confirming your payment…","confirmingDescription":"A few seconds: your subscription starts as soon as the bank has answered.","sepaManagedTitle":"Your subscription is paid by direct debit","sepaManagedDescription":"Payment is collected by direct debit from your bank account: there is nothing for you to do.","notApplicableTitle":"Nothing to pay right now","notApplicableDescription":"There is no renewal to complete from this link. Check your account for the current state of your subscription.","notApplicableActiveUntil":"Your subscription is running until {date}. There is nothing to renew from this link.","invalidTitle":"This link is not valid","invalidDescription":"This renewal link is incomplete or no longer exists. Sign in to your account to renew your subscription.","expiredTitle":"This link has expired","expiredDescription":"This renewal link is no longer valid. Sign in to your account to renew your subscription, or browse our offers.","errorTitle":"Something went wrong","errorDescription":"We could not load your summary right now. Please try again in a moment.","retryCta":"Try again","goToAccountCta":"Go to my account","seeOffersCta":"See our offers","paymentLinkError":"We could not retrieve the payment link. Please try again.","linkAlreadyUsedToast":"This link has already been used. Here is the current state of your subscription.","usedTitle":"This link has already been used","usedDescription":"Your request was already processed on a previous visit. See the current state of your subscription in your account.","tooManyAttempts":"Too many attempts. Please wait a minute before trying again.","notRenewableToast":"This renewal can no longer be completed from this link. Here is the current state of your subscription.","expiresOn":"This link stays valid until {date}.","reloadedTitle":"Reopen the link from your email","reloadedDescription":"For your security, the renewal link is removed from the address bar as soon as it opens. Reopen it from your email to continue, or sign in to your account.","gift":{"cancelled":{"giftUnavailableDescription":"We could not award this gift. Your renewal went through and your subscription is up to date.","giftUnavailableTitle":"Gift not awarded","nothingOwedDescription":"This gift came with a payment. Nothing was due for this operation, so the gift was not awarded. Your subscription is up to date.","nothingOwedTitle":"Gift not awarded","offerWithdrawnDescription":"This gift offer is no longer available. Your renewal is unaffected.","offerWithdrawnTitle":"Gift offer withdrawn","paidElsewhereDescription":"Your payment was received, but it was settled outside the online journey, so this gift cannot be awarded automatically. Get in touch if you believe you are entitled to it.","paidElsewhereTitle":"Gift not awarded","paymentIncompleteDescription":"The associated payment did not go through, so the gift was not awarded.","paymentIncompleteTitle":"Gift not awarded"},"description":"Choose your gift for this renewal.","errors":{"declineFailedContinuing":"We could not cancel your gift. Your renewal is going ahead, and the gift already saved may still be sent to you.","droppedContinuing":"This gift could not be saved. We are continuing with your renewal without it.","generic":"Unable to save your selection. Please try again.","locked":"This gift can no longer be changed. Reload the page.","required":"Complete your selection before continuing.","reserved":"A gift is already reserved for this renewal. Reload the page to see its status.","stale":"This offer has been updated. Reload the page to see the available gifts.","savedGiftLocked":"Your gift has already been saved for this renewal and can no longer be cancelled from this page. Pick at least one cover to continue, or get in touch."},"issueNumber":"No. {number}","loadError":"Unable to load your gift right now.","paidContactDescription":"We're preparing your gift for shipping. Our team will contact you to finalize delivery.","paidDescription":"We're preparing your gift for shipping. There's nothing else for you to do.","paidTitle":"Your gift is confirmed","pendingDescription":"Your selection is saved. Complete payment to confirm your gift.","pendingTitle":"Payment pending","resumePayment":"Resume payment","declineOption":"No thanks","selectionBounds":"{min, plural, =0 {Up to {max, plural, one {# choice} other {# choices}}} other {{min} to {max} choices}}","summaryTitle":"Your gift selection","summaryEmpty":"Pick your gift from the covers on offer — it will show up here.","summaryNote":"Your gift will be shipped as soon as your renewal is confirmed.","summaryIncluded":"Included","removeOption":"Remove “{title}” from my selection","removeBlocked":"This choice is required: pick another cover to replace it.","payCta":"Renew now","payWithoutGiftCta":"Renew without a gift","stickyPayCta":"{label} — payment bar","conditions":"Offer limited to renewals paid online from this link, while stocks last.","stickySelected":"{count, plural, =0 {No gift} one {# gift selected} other {# gifts selected}}","previousPage":"Show the previous covers","nextPage":"Show the next covers","pageIndicator":"{page} / {total}","pageStatus":"Page {page} of {total}"}},"etablissement":{"portal":{"summary":{"statusPending":"Awaiting subscription","statusActive":"Active subscription","statusInactive":"Inactive subscription","statusArchived":"Archived","seatsLabel":"Seats","seatsUsed":"{seatCount} of {maxSeats} used","checkoutCta":"Complete my information and pay"},"noEtablissementTitle":"No institution linked","noEtablissementDescription":"Your account is not linked to an institution. If you are a teacher, submit your request from the dedicated area.","requestCta":"Submit a request","errorTitle":"Error","errorDescription":"Your institution could not be loaded.","members":{"title":"Teachers","seatsUsed":"{seatCount} of {maxSeats} seats used","add":"Add a teacher","seatsFull":"You have reached the limit of {maxSeats} seats. Deactivate a seat to free one up.","emptyTitle":"No teachers yet","emptyHint":"Click \"Add a teacher\" to invite a colleague.","deactivate":"Deactivate","deactivateConfirmTitle":"Deactivate this teacher's seat?","deactivateConfirmDescription":"Their digital access is revoked immediately. The seat can then be reassigned to another teacher.","deactivateError":"The seat could not be deactivated.","seatDisabledByAdmin":"Disabled by admin","seatActive":"Active seat","seatClosed":"Closed seat","inviteTitle":"Invite a teacher","inviteDescription":"The teacher receives an invitation email. If an account already exists, it is linked straight away.","emailLabel":"Work email","emailPlaceholder":"first.last@school.org","errors":{"alreadyActiveElsewhere":"This teacher already holds an active seat in another institution. Ask them to leave their current institution before inviting them here.","seatCapExceeded":"Seat limit reached. Deactivate a seat before adding another one.","inviteFailed":"This teacher could not be invited. Check the email address, then try again."}}},"signup":{"howItWorks":{"title":"Two steps, one minute.","oneMinute":"One minute","instruction":"Click « Get my free subscription » to send your request to your school library. We'll take care of the rest.","step1Title":"Enter your details","step1Description":"Last name, first name and email to activate your free unlimited digital access.","step1Time":"40 seconds","step2Title":"Forward the confirmation email to your school library","step2Description":"Simply provide the email of your school library or librarian teacher.","step2Time":"20 seconds"},"cta":{"getFreeSubscription":"Get my free subscription"},"hero":{"badge":"Teachers' Space","titleLead":"Free digital","titleFree":"access","titleFor":"for","titleTeachers":"Teachers","description":"Fill in a single form and immediately receive your Tangente purchase order, ready to send to your school library or administration. As soon as the subscription is validated, your digital access is activated automatically, for free, for 1 year.","imageAlt":"Tangente Numérique and podcast cover — teachers' offer","archivesYears":"12 years","archivesLabel":"of archives"},"wizard":{"errors":{"duplicate":"A request for this teacher already exists. Contact us if needed.","generic":"Unable to send the request. Check your connection and try again."},"title":"Receive your Tangente purchase order","description":"Enter your details and the name of your school. We'll email you a purchase order ready to send to your school library or administration.","firstNamePlaceholder":"Camille","lastNamePlaceholder":"Martin","emailLabel":"Work email","emailPlaceholder":"firstname.lastname@school.com","emailDescription":"You will receive your purchase order and account activation at this address.","etablissementLabel":"School name","etablissementPlaceholder":"Jules Ferry High School","etablissementDescription":"The purchase order will be pre-filled with your school's name.","postalCodePlaceholder":"75001","townPlaceholder":"Paris","submitCta":"Receive the purchase order by email","trustEncrypted":"Encrypted data","trustNoCommitment":"No commitment on your part","trustEmailSent":"Purchase order sent by email"},"help":{"title":"A question before you get started?","description":"Our team supports every school subscription request. Purchase orders, administrative mandates, invoices: we adapt to your bursar's process.","emailLabel":"Write to us","phoneLabel":"Call us","phoneValue":"{phone} — Monday to Friday"},"footer":{"legalNotice":"Legal Notice","termsOfSale":"Terms of Sale","privacy":"Privacy"},"success":{"title":"A confirmation email has just been sent to you.","whatNext":"What now?","step1Title":"Check your spam folder","step1Description":"the email may have been filtered by mistake.","step2Title":"Simply forward the email to your school library","step2Description":"directly.","done":"That's it!"}}},"seo":{"articles":{"title":"Articles","description":"All Tangente Magazine articles.","pageTitle":"Articles - Page {page}"},"authors":{"title":"Authors","description":"Discover Tangente's authors — biographies, published articles and links.","notFoundTitle":"Author","pageTitle":"Author - Page {page}"},"categories":{"fallbackTitle":"Category","fallbackDescription":"Mathematics articles by category.","descriptionWithName":"Articles in the {name} category.","pageTitle":"{name} - Page {page}"},"glossaire":{"title":"Glossary","description":"Find every mathematical definition from Tangente, sorted alphabetically.","letterTitle":"Glossary: letter {letter}","letterDescription":"Tangente's mathematical definitions starting with the letter {letter}.","lettersHeading":"Browse by letter","letterLinkLabel":"Letter {letter}, {count, plural, one {# term} other {# terms}}"},"contact":{"description":"Contact the Tangente Magazine team by email or phone.","title":"Contact"},"contactConfirmation":{"description":"Confirmation of your subscription to the Tangente newsletter.","title":"Subscription confirmation"},"dossierDetail":{"fallbackTitle":"Folder"},"dossiers":{"description":"Explore our themed folders on mathematics.","pageTitle":"Folders - Page {page}","title":"Folders"},"home":{"description":"Tangente Magazine - The accessible mathematics magazine. Discover our articles, folders and issues."},"magazineDetail":{"fallbackTitle":"Magazine"},"magazines":{"description":"All issues of Tangente Magazine - The accessible mathematics magazine.","pageTitle":"Magazines - Page {page}","title":"Magazines"},"mathematicalThemeDetail":{"description":"Explore mathematics articles on the theme of {theme}.","title":"Theme: {theme}","pageTitle":"Theme: {theme} - Page {page}"},"mathematicalThemes":{"description":"Explore mathematics articles by theme: geometry, algebra, analysis, arithmetic and more.","themeLabel":{"algebra":"Algebra","analysis":"Analysis","applied_mathematics":"Applied mathematics","arithmetic":"Arithmetic","combinatorics_and_games":"Combinatorics and games","geometry":"Geometry","logic":"Logic","numerical":"Numerical","probability_and_statistics":"Probability and statistics"},"title":"Mathematical themes","pageTitle":"Mathematical themes - Page {page}"},"notFound":{"title":"Page not found | Tangente Magazine"},"privacyPolicy":{"title":"Privacy Policy","description":"Tangente Magazine privacy policy."},"termsOfSale":{"title":"Terms of Sale","description":"Tangente Magazine terms of sale."},"legalNotice":{"title":"Legal Notice","description":"Tangente Magazine legal notice."},"aPropos":{"title":"About","description":"Discover Tangente, the reference magazine for popularizing mathematics, published by Éditions POLE since 1987."},"faq":{"title":"Frequently Asked Questions","description":"Find answers to your questions about subscriptions, payment, content access and delivery for Tangente Magazine."},"olympiadesMathematiques":{"title":"Mathematics Olympiad","description":"Tangente is offering you 3 months of free digital subscription."},"espaceProfesseurs":{"title":"Free Tangente Digital for mathematics teachers","description":"Mathematics teachers: get free access to Tangente Digital + Podcast (100+ issues, 12 years of archives, 500+ problems). Request your school's print subscription in 1 minute — your CDI takes care of the rest."},"espaceProfesseursDemande":{"title":"School Subscription Request — Teachers' Space","description":"Submit your Tangente subscription request to your school's CDI or bursar's office — Tangente Digital + Podcast access offered free for 1 year to the teacher."},"pricing":{"title":"Subscription","description":"Discover our Tangente Magazine subscription plans."},"pricingInformations":{"title":"Information & Payment"},"problem":{"fallbackTitle":"Problem"},"site":{"title":"Tangente Magazine","description":"Tangente Magazine - the magazine for accessible mathematics."}},"authors":{"shareCard":{"title":"Share this author profile","copyLink":"Copy link","linkCopiedAria":"Link copied","linkCopiedText":"Link copied!"},"otherAuthorsCard":{"title":"In the same section"},"socialCard":{"title":"Social media"},"articlesPagination":{"ariaLabel":"Pagination","previousPage":"Previous page","nextPage":"Next page"},"articlesSection":{"eyebrow":"Signed articles","articleWord":"{count, plural, one {article} other {articles}}","publishedIn":"{count, plural, one {published in Tangente} other {published in Tangente}}","empty":"No articles published yet."},"verifiedBadge":"Verified author","readArticle":"Read →","fallbackName":"Author","detailTitleSuffix":"Author","title":"Authors","countLabel":"{count, plural, =0 {{count} authors listed.} one {{count} author listed.} other {{count} authors listed.}}"},"contact":{"confirmationBackHome":"Back to home","confirmationInvalidDetail":"This confirmation link is no longer valid. It may have already been used, or it may have expired (links are valid for 7 days).","confirmationInvalidRetry":"Sign up again from the site to receive a new link.","confirmationInvalidTitle":"Invalid or expired link","confirmationSuccessDetail":"You will receive our upcoming publications directly in your inbox.","confirmationSuccessGeneric":"Your newsletter subscription is now active.","confirmationSuccessTitle":"Thank you, your subscription is confirmed!","confirmationSuccessWithPerimeter":"You are now subscribed to the {perimeter} newsletter.","emailLabel":"Email","heading":"Contact us","phoneLabel":"Phone","subheading":"Our team is available to answer your questions."},"dossiers":{"detail":{"articlesHeadingPrimary":"All articles ","headingSecondary":"in this folder","notTranslatedNotice":"This folder is not yet available in English; the French version is shown.","partlyTranslatedNotice":"Some articles in this folder are not yet available in English and are shown in French."},"card":{"viewFull":"View the full folder"}},"magazines":{"detail":{"articlesHeadingPrimary":"All articles ","buyPaperVersion":"Buy the print edition","foldersHeadingPrimary":"All folders ","thisIssueSuffix":"in this issue","notTranslatedNotice":"This issue is not yet available in English; the French version is shown.","partlyTranslatedNotice":"Some content in this issue is not yet available in English and is shown in French.","expectedRelease":"Expected on {date}"},"inStores":"On sale now"},"notFound":{"backHome":"Back to home","browseArticles":"Browse articles","description":"The address you’re following no longer exists, has moved, or never existed. Let’s pick up the thread — there’s plenty to explore on the mathematics side.","errorLabel":"Error 404","figcaption":"A tangent, by definition, never stops anywhere.","figureLabel":"fig. 404","heading":"This page has wandered off the tangent line.","linkContact":"Contact us","linkFolders":"Folders","linkMagazines":"Magazines","linkThemes":"Mathematical themes","orElse":"Or","quickLinksAriaLabel":"Quick links","svgAriaLabel":"A tangent line that never meets its curve","svgFormulaLine1":"lim x → page","svgFormulaLine2":"= not found","simpleTitle":"Page not found","simpleDescription":"The page you are looking for does not exist or has been moved."},"faq":{"title":"Frequently asked questions","subtitle":"Find answers to the most common questions about Tangente Magazine.","categories":{"0":{"title":"Subscription & Pricing"},"1":{"title":"Payment & Billing"},"2":{"title":"Accessing content"},"3":{"title":"My account"},"4":{"title":"Delivery"},"5":{"title":"Contact & Support"},"6":{"title":"Anthology offer"}},"items":{"0":{"question":"What is the difference between Digital and Paper + digital?","answer":"**Digital** gives you access to every article, dossier and issue on tangente-mag.com and the app (coming soon), without receiving the print magazine.\n\n**Paper + digital** adds home delivery of issues, special editions and « Bibliothèque » titles. Digital access is included — it's the complete package."},"1":{"question":"How many magazines will I receive with Paper + digital?","answer":"It depends on the plan you choose:\n\n**HALF-YEARLY SUBSCRIPTIONS**\n- **Plus** — 3 issues + 2 Kiosque special editions per half-year\n- **Super Plus** — 3 issues + 2 Bibliothèque Tangente per half-year\n- **Soutien** — 3 issues + 2 special editions + 2 Bibliothèque Tangente per half-year\n\n**YEARLY SUBSCRIPTIONS**\n- **Plus** — 6 issues + 4 Kiosque special editions per year\n- **Super Plus** — 6 issues + 4 Bibliothèque Tangente per year\n- **Soutien** — 6 issues + 4 special editions + 4 Bibliothèque Tangente per year"},"2":{"question":"How does the €1 first month offer work?","answer":"The monthly Digital offer starts at €1 for the first month — ideal for trying Tangente.\n\nThe price then automatically switches to €5.90/month.\n\nYou can cancel at any time from your personal account — including during the first month, in one click."},"3":{"question":"Can I cancel my subscription?","answer":"Yes, free of charge and with no justification needed.\n\nFor every plan, cancelling your subscription switches off automatic renewal. You keep access to the web and/or paper service until the end of the period you already paid for (monthly, half-yearly or yearly, depending on your plan)."},"4":{"question":"What subscription plans are available?","answer":"Tangente Magazine offers **digital** subscriptions (online access only) and **paper + digital** subscriptions (the magazine delivered by post plus online access). Each plan is available for several durations: monthly, half-yearly, yearly or two-yearly. The longer the duration, the better the price."},"5":{"question":"How can I subscribe?","answer":"Go to the [Pricing](/pricing) page to see our offers. Pick the plan that suits you, configure the options (duration, delivery), then pay online. You will need to create an account if you don't already have one."},"6":{"question":"Can I change plans during my subscription?","answer":"Changing plans mid-subscription is not available as a self-service option yet. If you would like to change your plan, contact our support team by email at **abo@poleditions.com** or by phone on **02 32 22 13 93** and we'll take care of it."},"7":{"question":"What is automatic renewal?","answer":"Automatic renewal extends your subscription at the end of each period with no interruption of service. You can choose to turn this option on or off when you subscribe, on the [Pricing](/pricing) page. Once it's on, if you change your mind, you can cancel your subscription from **My account > Subscription**."},"8":{"question":"Do you offer refunds?","answer":"Digital subscriptions are non-refundable once activated. For paper subscriptions, contact our support team at **abo@poleditions.com** or on **02 32 22 13 93** so we can look into your situation."},"9":{"question":"What payment methods do you accept?","answer":"We accept payments by **credit card** (Visa, Mastercard, American Express) through our secure payment platform, Stripe."},"10":{"question":"What should I do if a payment fails?","answer":"If your payment fails, you'll receive a notification. Go to **My account > Subscription**, where a button lets you retry the payment. Check that your card is valid and that you haven't reached your spending limit."},"11":{"question":"How do I update my card?","answer":"From **My account > Subscription**, click **Change payment method**. You'll be redirected to a secure form to enter your new card details."},"12":{"question":"Where can I find my invoices?","answer":"All your invoices are available under **My account > Invoices**. You can view and download them at any time."},"13":{"question":"Do the prices include delivery fees?","answer":"Delivery within **mainland France** is included in the price of the paper subscription. For deliveries **outside mainland France** (overseas territories, international), a surcharge applies and is shown while you set up your subscription."},"14":{"question":"What content is free to access?","answer":"You can freely browse the titles, descriptions and previews of every article. Full access to article and issue content is reserved for subscribers."},"15":{"question":"How do access quotas work?","answer":"Depending on your plan, you have a quota of articles you can read per billing period. Your usage is shown under **My account > Subscription**. Quotas reset at the start of each new period."},"16":{"question":"Can I access back issues of the magazine?","answer":"Yes, subscribers have access to every issue and article published on the platform, within the limit of their quota."},"17":{"question":"What is the difference between an article, an issue and a dossier?","answer":"An **article** is a single piece of writing covering one mathematical topic. An **issue** (or magazine) is a complete edition of Tangente containing several articles organised into sections. A **dossier** is a thematic collection of articles built around one specific subject."},"18":{"question":"How often is content updated?","answer":"New articles are published on the platform on a regular basis. Issues of Tangente magazine come out according to the usual editorial schedule."},"19":{"question":"How do I create an account?","answer":"Click **Log in** at the top of the page, then follow the instructions to create your account. You can also create an account while subscribing."},"20":{"question":"I've forgotten my password — what should I do?","answer":"On the login page, click **Forgot password**. You'll receive an email with a link to reset your password."},"21":{"question":"How do I manage my delivery addresses?","answer":"From **My account > Addresses**, you can add, edit or delete the delivery addresses for your paper subscriptions."},"22":{"question":"Do you deliver abroad?","answer":"Yes. Delivery is included in the subscription price for mainland France.\n\nFor Belgium, Switzerland and the rest of the world, extra fees apply and are shown at checkout. Digital access, on the other hand, is exactly the same everywhere."},"23":{"question":"What are the delivery times?","answer":"Magazines are shipped as soon as they're released. Allow roughly **5 to 10 business days** for mainland France and **2 to 4 weeks** internationally, depending on the destination."},"24":{"question":"Can I change my delivery address during my subscription?","answer":"Yes, you can change your delivery address at any time from **My account > Addresses**. The change will apply to the next shipment."},"25":{"question":"I haven't received my magazine — what should I do?","answer":"If you haven't received your magazine within the times shown above, contact us at **abo@poleditions.com** or on **02 32 22 13 93**, stating your subscriber number and delivery address. We'll do what's needed to sort out the problem."},"26":{"question":"How do I contact support?","answer":"You can reach us by email at **abo@poleditions.com** or by phone on **02 32 22 13 93**. We do our best to reply as quickly as possible."},"27":{"question":"How do I report a problem on an article?","answer":"On every article, a floating **Report a problem** button lets you flag an error or an issue straight from the article page. You need to be logged in to your account to use this feature."},"28":{"question":"How do I report a technical problem?","answer":"If you run into a technical problem (a page that won't load, a payment error, etc.), contact our support team at **abo@poleditions.com** and describe the problem as precisely as you can. Screenshots are welcome — they help us fix the issue quickly."},"29":{"question":"When does my anthology arrive?","answer":"The parcel ships within 72 working hours of payment, before your first subscription issue."},"30":{"question":"Can I choose the titles?","answer":"No: each anthology is put together in advance. That is what lets us ship it right away."},"31":{"question":"What if a title is out of print?","answer":"We replace it with an equivalent issue of at least the same value. The composition shipped appears on your order slip."},"32":{"question":"Does the offer apply to the 100% digital plan?","answer":"No: the anthology is a paper parcel. It comes with the Paper + digital plans only."},"33":{"question":"What if I only want to commit for six months?","answer":"The six-month subscription is still available on our usual subscription page. The anthology starts at one year: below that, shipping the parcel would cost more than the subscription brings in."},"34":{"question":"Can this link be reused?","answer":"This link is personal and valid for 45 days. Once your subscription is paid, it no longer serves."}}},"article":{"notTranslatedNotice":"This article is not yet available in English; the French version is shown.","readIn":"Read in: {language}","shopBook":{"kicker":"The bookshop","onShop":"In the bookshop","shopName":"Infinimath","buy":"Buy the book","opensInNewTab":"(opens in a new tab)"}},"articleComments":{"title":"Comments","count":"{count, plural, =0 {No comments} one {# comment} other {# comments}}","guidelines":"Please stay courteous. Comments are reviewed before publication.","empty":"No comments yet. Be the first to respond.","closed":"Comments are closed on this article.","composer":{"label":"Your comment","replyLabel":"Your reply to @{handle}","placeholder":"Join the discussion…","replyPlaceholder":"Reply to @{handle}…","hint":"Plain text; LaTeX formulas between $…$. 2,000 characters max.","remaining":"{count, plural, one {# character left} other {# characters left}}","submit":"Post","submitReply":"Reply","cancel":"Cancel"},"signIn":{"message":"Sign in to post a comment and talk with other readers.","action":"Sign in"},"invitation":{"title":"Join the discussion"},"verifyEmail":{"message":"Confirm your email address to comment."},"subscribe":{"message":"Comments on this article are reserved for digital subscribers."},"challenge":{"verifying":"Security check in progress…","loadFailed":"The security check could not be loaded."},"pending":{"title":"Awaiting review","body":"Your comment will be published once it has been reviewed.","badge":"Pending","loading":"Loading your comments…"},"posted":"Your comment has been published.","error":{"rateLimited":"You are posting too quickly. Wait a few minutes before trying again.","duplicate":"You have already posted this comment.","challengeFailed":"The security check failed. Please try again.","tooLong":"Your comment is longer than 2,000 characters."},"reply":"Reply","repliesTo":"Replies to @{handle}","showReplies":"Show {count, plural, one {the reply} other {the # replies}}","hideReplies":"Hide {count, plural, one {the reply} other {the # replies}}","options":"Comment options","loadMore":"Show more comments","readMore":"Read more","readLess":"Show less","removed":{"byAuthor":"This comment was deleted by its author.","byModeration":"This comment was removed by moderation.","replyDisabled":"You cannot reply to a deleted comment."},"delete":{"action":"Delete","confirmTitle":"Delete this comment?","confirmBody":"This cannot be undone. Replies to this comment will disappear too.","done":"Your comment has been deleted."},"kicker":"Discussion","threadLabel":"Comment thread","threadAuthor":"Thread author","moderate":{"menu":"Moderation actions","reject":"Reject comment","spam":"Mark as spam","skipped":"This comment has already been handled by another moderator.","done":{"rejected":"The comment has been rejected.","spam":"The comment has been marked as spam."}},"postedAt":"Posted on {date}"},"offer":{"strip":{"eyebrow":"New subscriber offer","countdown":"{days} d {hours} h {minutes} min {seconds} s left","headline":"The Tangente anthology, free","until":"With any new subscription until {date}","over":"Offer over","left":"Only","unit":{"days":"{count, plural, one {day} other {days}}","hours":"hours","minutes":"min","seconds":"sec"},"sub":"With any new paper subscription"},"title":"Choose your Tangente subscription","subtitle":"Maths to read, listen to and play.","gift":{"tag":"Free with your subscription","shipped":"Shipped immediately","digital":"Digital access included with every paper subscription","pickPlan":"Choose your plan to see the gift that comes with it.","noneForPlan":"No gift comes with this plan on this term. Choose another plan or term to receive one.","myGift":"My gift anthology","count":"{count, plural, =0 {No title} one {# free title} other {# free titles}}","stockNote":"While stocks last: an out-of-print title is replaced by an equivalent one of at least the same value.","loadError":"Your gift could not be loaded right now.","forPlans":"Free with {names}","loading":"Loading…","title":"The Tangente anthology, free","promise":"Up to {titles, plural, one {# title} other {# titles}} free, worth {value} in bookshops.","promiseTitlesOnly":"Up to {titles, plural, one {# title} other {# titles}} free.","promiseGeneric":"With any new one- or two-year paper subscription.","seeCollection":"See the free collection","valueLabel":"Your gift's bookshop value:","priceSource":"Prices from librairie-infinimath.com.","more":"+{count}","ribbon":"Free","chooseFirst":"Choose your gift first, then confirm the payment.","withheld":"Not available","seeCollectionFor":"See the free collection for {term}"},"plans":{"title":"Choose your plan","subtitle":"Paper + digital, digital access always included.","choose":"Choose this plan","selected":"Selected","empty":"No plan is available right now. Please try again in a moment.","from":"from","chooseNamed":"Choose the {name} plan"},"duration":{"title":"Choose the length of your subscription","plan":"{name} plan","mostOffered":"Most free titles","perMonth":"that is {amount} / month","giftLine":"Your gift: the Tangente anthology,","giftValue":"worth {value}","giftTitles":"{count, plural, one {# free title} other {# free titles}}","noGift":"no gift on this term"},"back":"Back","continue":"Continue my order","address":{"title":"Your delivery address","subtitle":"Your anthology and your issues will be shipped there."},"summary":{"plan":"Subscription","shippingAbroad":"Delivery supplement outside metropolitan France","commitment":"{interval, select, annual {One-year commitment} biennial {Two-year commitment} other {{term} commitment}}","today":"today","toPay":"Total to pay","included":"VAT and delivery included","secure":"Secure payment","protected":"Your details are protected","ssl":"SSL encrypted","legal":"By continuing, you accept the terms of sale and the privacy policy.","pay":"Continue to payment","cannotSubscribe":"This account cannot take out a subscription. Write to us at abo@poleditions.com.","title":"Your order"},"upsell":{"link":"Switch to {term} and get even more gifts ({titles, plural, one {# title} other {# titles}}{current, plural, =0 {} other { instead of #}})","revert":"Back to {term}"},"payWithGiftCta":"Continue and receive my gift","guarantees":{"shipping":"Parcel shipped within 72 working hours","metropole":"Gift not available outside mainland France","cancel":"Cancel at any time"},"reassure":{"payment":"Secure payment","shipping":"Delivery included in France","cancel":"Cancel at any time","renewal":"Renews automatically"},"errors":{"priceRefused":"This plan is not available with this link for this address. Check the term and the delivery address, then try again.","generic":"The payment could not start. Please try again in a moment.","address":"Check your delivery address.","termNotAbroad":"This term is not offered outside metropolitan France. Choose another term.","billingAddress":"Check your billing address.","paymentInProgress":"A payment is already in progress for this link. Wait a moment, then try again.","summary":"Your offer could not be loaded right now."},"faq":{"title":"Your questions, our answers","lead":"The questions we are asked most often about Tangente subscriptions."},"per":{"annual":"/ year"},"term":{"monthly":"1 month","semester":"6 months","annual":"1 year","biennial":"2 years"},"sheet":{"sub":"{count, plural, one {# title} other {# titles}} for {plan} {term} — shipped before your first issue.","valueLabel":"Your gift's bookshop value","close":"Close"},"feature":{"tangente_mag":"{count, plural, one {# issue} other {# issues}} of Tangente in print / year","hors_serie":"{count} “kiosque” special issues in print / year","bibliotheque":"{count} Bibliothèque Tangente / year (160-page book)"}},"bookshopOffer":{"title":"Activate your free year","eyebrow":"Bookshop offer","subtitle":"Your bookshop purchase includes 1 year of Tangente. Simply choose how to retrieve your offer.","orderNumber":"Order no. {number}","footer":{"question":"A question?","rights":"Tangente Magazine © {year} — All rights reserved"},"code":{"kicker":"Immediate activation","title":"Find your offer","lead":"Enter the e-mail used for your order and the code you received by e-mail.","emailLabel":"Infinimath bookshop order e-mail","codeLabel":"Activation code","codePlaceholder":"XXXX-XXXX-XXXX","hint":"12 characters, with or without dashes.","whereTitle":"Where is my code?","whereBody":"It has 12 characters and sits in the e-mail sent after your purchase.","submit":"Find my offer"},"entry":{"label":"How to claim your offer","code":"I have a code","receipt":"I have a receipt"},"receipt":{"kicker":"Receipt check","title":"Send your proof of purchase","lead":"We check it within a few working days, then e-mail you your code.","emailLabel":"Your e-mail","emailHint":"Your code will be sent to this address.","fileLabel":"Photo or PDF of the receipt","filePick":"Add your receipt","fileHint":"JPG, PNG, WebP or PDF · 10 MB max.","fileTooLarge":"This file is over 10 MB. Shrink the photo or take a new one.","fileType":"This format is not accepted. Send a photo (JPEG, PNG, WebP) or a PDF.","optional":"Add optional details","bookshopLabel":"Bookshop","bookshopPlaceholder":"Bookshop name","dateLabel":"Purchase date","consent":"Your receipt is used only to verify the purchase, then it is deleted.","submit":"Send my receipt","uploading":"Uploading… {percent}%","sentKicker":"Receipt sent","thanksTitle":"Thank you, we are checking your receipt","thanksLead":"You will hear back at {email} within a few working days. If the receipt is validated, the e-mail carries your activation code.","thanksNote":"Remember to check your spam folder.","another":"Send another receipt"},"error":{"422":"The form is incomplete: check the e-mail address and the file.","3429":"Too many attempts. Wait a few minutes before trying again.","8821":"Another customer already uses this e-mail address. Contact us to activate your offer.","9400":"The settings changed in the meantime. Reload the page.","9401":"The campaign is incomplete and cannot be enabled.","9402":"No offer matches this address and code. Check your order e-mail and the code you received.","9403":"The activation deadline for this offer has passed.","9404":"This offer has already been used.","9405":"This offer is no longer valid.","9406":"This address already has a running subscription: the offer is for new subscribers.","9407":"The offer is not active at the moment.","9408":"This offer belongs to another account.","9409":"This plan is not part of the offer.","9410":"A delivery address is required for this plan.","9411":"This plan cannot be delivered to that country.","9412":"An activation is already in progress for this offer. Finish it or wait for its confirmation.","9413":"This plan is no longer on sale.","9414":"The activation could not be started. Please try again.","9415":"A receipt is already being checked for this address. You will hear back by e-mail.","9416":"This address already holds an offer. Use the code you received by e-mail.","9417":"The anti-robot check failed. Please try again.","9418":"This receipt claim could not be found.","9419":"This receipt claim has already been decided.","9420":"The chosen threshold is not part of the campaign settings.","generic":"The activation could not go through. Try again in a moment."},"formule":{"title":"Choose your plan"},"planType":{"digital":"Digital plan","paper":"Print plan"},"address":{"title":"Delivery address","lead":"Your magazines will be posted to this address.","abroadSupplement":"Outside metropolitan France, a shipping supplement of {amount} a year is added at renewal. The first year stays free, shipping included."},"recap":{"title":"Activation","lead":"Check your plan, complete your account with the order e-mail, then activate your free year.","order":"Order","today":"Due today","freeUntil":"Free until","thenYearly":"Then","perYear":"{price} / year","terms":"A payment card is saved for later: the subscription renews automatically at the price shown when the {days} free days end, unless you cancel before that date from your account.","welcomeMail":"A new account also receives the usual welcome e-mail, on top of the offer confirmation.","activate":"Activate my free year"},"wrongAccount":{"title":"This account is not the order’s","lead":"The offer is tied to the order’s e-mail address. Sign out, then sign in or create an account with that address.","signOut":"Sign me out"},"success":{"missingTitle":"Offer not found","missingLead":"This page is reached from the offer payment. Start the activation again with your code.","backToOffer":"Back to activation","pendingTitle":"Activation in progress","pendingLead":"We are confirming your free year, this takes a few seconds…","stillPending":"The confirmation is taking longer than expected. You will receive an e-mail as soon as your free year is active.","signInLead":"Sign in with your order’s address to follow the activation.","signIn":"Sign in","readyTitle":"Your free year is active!","readyLead":"A confirmation e-mail has been sent to you. Happy reading!","checkAgain":"Check again"}}},"timeZone":"Europe/Paris","children":"$L22"}]}]]}] 14:null 18:[["$","title","0",{"children":"Articles - Page 110 | Tangente Magazine"}],["$","meta","1",{"name":"description","content":"All Tangente Magazine articles."}],["$","link","2",{"rel":"canonical","href":"https://tangente-mag.com/en/articles/page/110"}],["$","meta","3",{"name":"format-detection","content":"telephone=no, date=no, address=no, email=no"}],["$","meta","4",{"property":"og:title","content":"Articles - Page 110 | Tangente Magazine"}],["$","meta","5",{"property":"og:description","content":"All Tangente Magazine articles."}],["$","meta","6",{"property":"og:url","content":"https://tangente-mag.com/en/articles/page/110"}],["$","meta","7",{"property":"og:site_name","content":"Tangente Magazine"}],["$","meta","8",{"property":"og:locale","content":"en_US"}],["$","meta","9",{"property":"og:type","content":"website"}],["$","meta","10",{"name":"twitter:card","content":"summary"}],["$","meta","11",{"name":"twitter:site","content":"@TangenteMag"}],["$","meta","12",{"name":"twitter:title","content":"Articles - Page 110 | Tangente Magazine"}],["$","meta","13",{"name":"twitter:description","content":"All Tangente Magazine articles."}],["$","link","14",{"rel":"icon","href":"/favicon.ico"}],["$","link","15",{"rel":"apple-touch-icon","href":"/logo-tangente-icon.jpg"}],["$","$L23","16",{}]]

All articles

Prime avoidance lemma (algebra) | Tangente

Mutually singular measures and Lebesgue decomposition | Tangente

Coprime (comaximal) ideals in algebra | Tangente

Simple function and Lebesgue integral | Tangente

Étale morphism in algebra and geometry | Tangente

Equiprojective: screws and rigid-body mechanics | Tangente

Equiprobability: Laplace's rule explained | Tangente

Equipotence: Bijections and infinite cardinalities | Tangente

Equidistant: locus and perpendicular bisector | Tangente

Equipollence: free vectors and directed segments | Tangente

Equicontinuity and the Arzelà–Ascoli theorem | Tangente

Uniformly integrable random variables | Tangente

Norm equations in algebra | Tangente

Quadratic equation: the discriminant | Tangente

Difference equation: sequences and methods | Tangente

Characteristic equation: eigenvalues | Tangente

Punctured neighborhood: definition and limits | Tangente

Epigraph of a convex function in analysis | Tangente

Comparable elements in a partial order | Tangente

Primitive element theorem in algebra | Tangente