The French word "discrépance" is uncommon. Derived from the Latin (discrepantia), it can figuratively mean "disagreement" or "discordance." The mathematical discrepancy problem is difficult, but it can be presented in elementary, metaphorical terms (see Tangente 168, 2016). It can also be approached as a game. Let us see what the mathematical discrepancy problem looks like in this setting.
The placer and the collector
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The game we shall consider is played by two people known, for reasons that will soon become clear, as the placer and the collector. Initially, the placer has an unlimited supply of identical marbles in each of two colours, white and green. He walks in a straight line, laying a marble on the ground after every metre. Each time, however, he must choose the colour "carefully," except for the first marble, which by convention is white.
The collector follows him from the same starting point, picking up marbles and placing them on one of the two pans of a balance: white marbles on the right pan and green ones on the left. He may choose to collect a marble every metre, every two metres, every three metres, and so on. Moreover, whatever interval he chooses, the balance must remain "properly balanced," meaning that the numbers of marbles on the two pans may differ by at most one.
The placer's aim is to lay as many marbles as possible. At first glance, the game may seem easy: surely the placer can simply alternate white and green marbles indefinitely. This works if the collector takes a marble every metre, but fails if he doubles the interval. A more subtle solution is therefore needed.
