72 Chapter 7

Did you find a winning strategy? Perhaps one might hope to discover how to win by

working backward from the winning condition. Imagine what number you must say on

your penultimate turn, the number from which your opponent cannot win, but you will be

able to win right after that. This would be the number 17, of course, since if you say 17,

then your opponent cannot quite reach 21, but whether your opponent ends on 18, 19, or

20, you will then be able to reach 21 and thereby win. By working further back from this,

we identify the winning strategy.

Theorem 49. The first player has a winning strategy in the game of Twenty-One. The

winning strategy is to end each turn with one of the numbers on this list:

1, 5, 9, 13, 17, 21.

Proof. That is, the first player should always stop counting on each turn exactly at one of

the numbers on the list. Notice that the first player can certainly start with a number on

the list, simply by saying the number one. And since the numbers on the list are spaced

four apart, whenever the first player has stopped at a number on the list, the opponent

cannot get up to the next number on the list, since the opponent can add only at most three

numbers. The key observation to make is that, because the opponent must add at least one

on each turn, it follows that the first player will thereby come within striking range of the

next number on the list, enabled to climb to it on the next turn. So the first player will

ultimately be able to say “twenty-one” and therefore win.

I would like to emphasize the uniqueness aspect of the winning strategy: we had referred

to the winning strategy, rather than a winning strategy. What I claim is that the strategy

we identified is in fact the only winning strategy, for if a player should ever fail to follow

the strategy, by saying a number not on the list, then the argument of the proof shows that

the opponent can immediately seize control of the game and take up the strategy simply by

climbing to the next number on the list.

The game of Twenty-One generalizes naturally to many other versions, where we adopt

a different target number or allow a different step size at each turn. For example, perhaps

we aim to count up to thirty-five or some other number; or perhaps we allow each player

to announce up to five numbers on each turn, or perhaps only two. So we really have an

infinite parameterized scheme of games here to analyze. Let G

n,s

denote the version of the

game with target n and allowed step size s, meaning that the two players cooperate to count

to n, starting with one and each adding up to the next s numbers on each turn; whoever

gets to n wins. In this notation, the original game of Twenty-One is G

21,3

, with a target of

21 and step size 3. Kindly play a few rounds of the game G

31,5

with your partner. Can you

find a winning strategy? Can you generalize the idea behind the proof of theorem 49?

Interlude. . .