Games and Winning Strategies

The Game of 21

Rules:

1. Start at 0.
2. Two players take turns adding 1, 2, or 3 to the count.
3. The player who reaches 21 wins.

Winning Strategy

To ensure you reach 21, you must be the one to reach 17. Why? If you are at 17:

• Opponent adds 1 $\to$ 18. You add 3 $\to$ 21.
• Opponent adds 2 $\to$ 19. You add 2 $\to$ 21.
• Opponent adds 3 $\to$ 20. You add 1 $\to$ 21.

To reach 17, you must reach 13. To reach 13, you must reach 9. To reach 9, you must reach 5. To reach 5, you must reach 1.

Strategy:

• Go first.
• Say 1.
• Whatever the opponent adds ($x$), you add ($4-x$).
• This ensures you land on multiples of 4 plus 1: $1, 5, 9, 13, 17, 21$.

General Rule

If the target is $T$ and you can add numbers $1$ to $k$: The winning positions are multiples of $(k+1)$.

• If $T$ is a multiple of $(k+1)$, the second player wins (if they play perfectly).
• If $T$ is not a multiple, the first player wins by taking the remainder of $T \div (k+1)$.