Problem
COM-B1-M08-P018 Moves \(2,3,5\)
#18
★★★☆☆ Level 3 of 5
There are \(52\) stones. In one move, a player may take \(2\), \(3\), or \(5\) stones. A player who cannot move loses. Find a winning first move.
The losing residues modulo \(7\) are \(0\) and \(1\).
Positions with residues \(0\) and \(1\) modulo \(7\) are losing: moves \(2,3,5\) from them do not lead back to residues \(0\) or \(1\). From residue \(2\) take \(2\), from \(3\) take \(2\), from \(4\) take \(3\), from \(5\) take \(5\), and from \(6\) take \(5\), leaving \(0\) or \(1\). Since \(52\equiv3\pmod7\), the first player takes \(2\) stones and leaves \(50\equiv1\pmod7\).
Not every game has move \(1\), so the terminal positions matter.