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How to Play the Twenty-One Game
The Twenty-One Game is a mathematical strategy game where players take turns adding numbers to a running total. The player who reaches the target number exactly wins.
Game Rules:
- Players take turns adding a number (1 to the maximum per turn) to the running total
- The total starts at 0
- The first player to reach exactly the target number (21 in classic play) wins
- If a player causes the total to exceed the target, they lose
Winning Strategy:
The Twenty-One Game has a perfect mathematical strategy based on modular arithmetic and the concept of "winning positions".
For a target of 21 and maximum number of 3 per turn, the winning positions are:
Where n = 0, 1, 2, 3, 4, 5, ...
For classic play (target 21, max 3), the winning positions are:
21, 17, 13, 9, 5, 1
If you can end your turn on one of these numbers, and play perfectly afterward, you will win.
Game Variations:
You can change the game by modifying:
- Target Number: The final number players are trying to reach
- Maximum Number Per Turn: The largest number a player can add on a single turn
Each variation has its own set of winning positions based on the formula above.
Mathematical Connection:
This game demonstrates concepts from combinatorial game theory and number theory. The winning strategy is based on modular arithmetic, specifically finding numbers that are congruent to 1 modulo (max+1).