For an appetizer, let us cut a salami into slices of varying thickness. One cut through the salami gives us two slices (particularly generous ones, admittedly). Two cuts give us three slices… and n cuts give us n + 1 slices. No surprises so far: this is clearly the way to obtain the greatest possible number of slices with n cuts, since, unless they have overindulged in the drinks on the table, nobody would dream of cutting twice in exactly the same place. Yet this question about cold cuts is the starting point for some less trivial enquiries.
Who wants a slice of pizza?
----------------------------
You may remember a certain famous, slightly plump Gaul who, when asked to cut three slices from a partly eaten cake intended for Cleopatra, managed it with just two strokes of his knife. Let us make matters a little more precise and consider dividing a pizza, or a disk, with n cuts—that is, using n lines.
One cut across the disk gives us two slices. Two cuts give us four. And what about three cuts?
The usual arrangement at family meals or evenings with friends produces six slices, but what is the maximum number we can obtain? The slices need not be equal at all (that is another, equally rich problem, though in practice some guests are always hungrier than others).
