Race to 21
The total starts at 0. You and the computer take turns to add 1, 2 or 3. Whoever makes the total exactly 21 wins. Tap a big button to play. The computer never makes a mistake, but it can be beaten. Lose a few games first, then open the secret box.
What's going on
Work backwards from the end. If you say 17, the other player must say 18, 19 or 20, and from each of those you can reach 21. So saying 17 is as good as winning.
Now 17 is your new target. If you say 13, the other player must say 14, 15 or 16, and you can reach 17 from each. Keep going back: 9, then 5, then 1. These are the safe numbers: 1, 5, 9, 13, 17, 21.
They go up in fours, and here is why. Whatever the other player adds, you add the number that makes the two steps total 4. They add 1, you add 3. They add 2, you add 2. They add 3, you add 1. You hop from one safe number to the next and they can never land on one.
So in the race to 21, go first and say 1. Then change the target or the biggest step and work out the new safe numbers before you open the box. Sometimes it is better to let the computer go first. When?
Sheet: The race to 21. Next year's game is Nim, which has a deeper secret.