Play the game to reach other target numbers (like 10, 11, or 12) by adding 1 or 2 each time. Can you find a number in each case when you are sure that you can win?
Target 10: Winning positions are 7, 4, and 1. ∴ If you reach 7, you can always win.
Target 11: Winning positions are 8, 5, and 2.
Target 12: Winning positions are 9, 6, and 3.
Key Strategy: The key is to reach a number that is 3 less than the target, then 3 less again, and so on.
Explanation
The chapter explains the NIM game strategy. The winning positions are found by working backwards from the target. For any target, subtract 3 repeatedly to find winning positions. This is because if you can reach a position where you're 3 away from the target, whatever your opponent adds (1 or 2), you can add the complement (2 or 1) to progress by exactly 3 and maintain control.
Solution Steps
For target 10: Winning positions are 7, 4, and 1
For target 11: Winning positions are 8, 5, and 2
For target 12: Winning positions are 9, 6, and 3
Key strategy: Reach a number 3 less than the target, then 3 less again