# 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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