Problem
COM-B1-M08-P007 The Last Stone Loses
#7
★★☆☆☆ Level 2 of 5
There are \(28\) stones. In one move, a player may take from \(1\) to \(3\) stones. The player who takes the last stone loses. Who wins?
With this rule, numbers \(1\pmod4\) are losing.
If one stone remains, the player to move loses. Then losing positions repeat every \(4\): \(1,5,9,\ldots\). Since \(28\equiv0\pmod4\), the first player takes \(3\) stones and leaves \(25\equiv1\pmod4\). Then the first player complements each opponent move to \(4\), until the opponent is left with one stone.
Contrast with the normal last-move-wins game.