Problem
COM-B1-M08-P021 Three Piles \(3,4,5\)
#21
★★★★☆ Level 4 of 5
There are three piles of \(3\), \(4\), and \(5\) stones. In one move, a player may take any positive number of stones from one pile. The last move wins. Find a winning first move and explain the continuing strategy.
Use the nim-sum \(3\oplus4\oplus5\).
Compute the nim-sum: \(3\oplus4\oplus5=2\). We need to make it \(0\). If the pile \(3\) is reduced to \(1\), then \(1\oplus4\oplus5=0\). After any opponent move, the nim-sum becomes nonzero; one can reduce one pile to make it \(0\) again. Therefore the winning move is to take \(2\) stones from the pile \(3\).
First full appearance of nim-sum.