Labels, the math enigma of “Le Monde” No. 67

When the game begins, the 10 candidates enter the room and sit silently on the 10 armchairs arranged in a circle. A sticker is then placed on each of their foreheads, on which is written a number from 1 to 10. The distribution of numbers is decided by the game master without any constraints: several players can wear the same number, and there may be numbers that are not used.

Each player sees the numbers written on the foreheads of the other 9 players, but does not see their own. Each of them must then secretly write a prediction on a slate about the number on their forehead. At the game master's signal, everyone turns over their slate at the same time. If at least one of the players makes the correct prediction—that is, the number written on their slate is the same as the one on their forehead—the game is won, and the 10 candidates, even those who were wrong, share the prize pool. But if they are all wrong, everyone leaves empty-handed.

What are the chances of candidates winning if they play randomly, without a strategy?

Once inside the room, the contestants are not allowed to communicate in any way, but they are allowed to consult with each other before the game to develop a common strategy. Is it possible for them to increase their chances of winning?

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