The game of 21

The game of 21, for two players. The first player says 1, 2 or 3. Then the players take turns adding 1, 2 or 3 to the number just said. The first to reach 21 wins!

The winning secret: work backwards from 21. If you say 17, your friend must say 18, 19 or 20, and then you can always reach 21. So 17 is a winning number. In the same way, if you say 13, you can always reach 17. Going back in fours, the winning numbers are 1, 5, 9, 13, 17, 21.

Why fours? Whatever your friend adds, you add whatever makes 4 in all: if they add 1, you add 3; if they add 2, you add 2; if they add 3, you add 1. So you hop from one winning number to the next. The first player starts by saying 1 and can always win.

A new game: start at 0, take turns adding 1, 2 or 3, and the first to reach 22 wins. Work back in fours from 22: 22, 18, 14, 10, 6, 2. So the first player should add 2 to reach 2, then keep saying 6, 10, 14, 18 and 22.