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Three Uneven Heaps

brainteaser
Three heaps of counters sit on a table. One heap contains 3 counters, another contains 4, and the last contains 5. You and an opponent alternate turns. On each turn, a player must choose exactly one heap and remove one or more counters from it. A player may empty a heap, but may not take counters from two different heaps during the same turn. The player who removes the final counter from the table wins. You move first, and your opponent will play perfectly. What should you remove on your opening turn to guarantee a win?

Hints

Hint 1
Hint 2
Hint 3

Solution

Commit an attempt to unseal the proof. Write your answer — right or wrong, thinking it through is the point.

What should you remove on your first turn?

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