Three piles of stones and the binary secret

Question

Three piles hold 3, 5, and 7 stones. Players alternate. On a turn you choose one pile and remove any positive number of stones from it. Whoever takes the last stone wins. Does the first player win, and what is the winning move?

Accepted Answer

The first player wins. The tool is the binary XOR (add the binary digits without carrying, keeping each column's parity). The losing positions for the mover are exactly those where this XOR is zero. Compute the XOR. Write the piles in binary: 3 = 011,\quad 5 = 101,\quad 7 = 111. Column by column (each column: is the count of 1s even or odd?): - Ones column: 1 + 1 + 1 = 3 → odd → 1 - Twos column: 1 + 0 + 1 = 2 → even → 0 - Fours column: 0 + 1 + 1 = 2 → even → 0 So the XOR is 001 = 1, which is not zero, the first player can win. Find the winning move. You must move to make the XOR zero. The non…

Three piles of stones and the binary secret

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