A Toothpick Game between Calvin and Brian!

Logic Level 4

Calvin and Brian play a game which begins with a pile of \(n\) toothpicks. They alternate turns with Calvin going first. On each player's turn, he must remove either 1,3 or 4 toothpicks from the pile. The player who removes the last toothpick wins the game.

Find the sum of the values of \(n\) from 31 to 35 inclusive for which Calvin has a winning strategy.

Bonus - Some interesting extensions: Can you figure out who has a winning strategy for \(n=100\)? Can you determine a complete list of winning positions for Calvin and Brian? What if, instead of removing 1,3 or 4, they remove 1,2 or 4. How about 1,3 or 6?

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