# The game of catching coins

My brother and I are very fond of the coin game. It consists of placing 20 coins on a table so that alternatively we are taking one, two or three coins, as desired by each player. The player who withdraws the last coins wins.

Is there any strategy to always win?

#### Solution

The winner is the last one who withdraws coins, that is, the first of the players with one, two or three coins left. Therefore, the first player with 4 chips will have lost since he must withdraw at least one coin and his opponent will have 1, 2 or 3 left so he can take them all to win the game.

Iteratively we could reason that any player who is with 8, 12, 16 or 20 chips will have lost (since the other player may in all cases reduce the number of chips to 4, 8, 12 or 16, respectively).

Thus, since there are 20 chips on the table in the picture, the first to withdraw chips is at a disadvantage and the second player can always apply the strategy to win, which should always strive to always leave a multiple number of chips on the table of 4.