The sticks game

The sticks game

In a game for two players, 23 sticks are placed on a table and each opponent will alternately take one, two or three sticks at a time, as they prefer. The player who is forced to take the last stick will lose.

What strategy can be followed to always win?


The player who leaves the last five sticks will win because his opponent must take 1, 2 or 3 and leave 4, 3 or 2 sticks respectively on the table so it will be enough to take 3, 2 or 1 to leave a single stick that will limp The other player

To leave 5 sticks, they must be left before 9.
To leave 9 sticks, they must be left before 13.
To leave 13 chopsticks, 17 must be left before.
To leave 17 sticks, 21 must be left.

So the starting player can ensure the win if he takes 2 sticks and follows the indicated rules.