Problem
COM-B1-M08-P020 Forty-Seven Stones
#20
★★★☆☆ Level 3 of 5
There are \(47\) stones. In one move, a player may take from \(1\) to \(4\) stones. The player who takes the last stone loses. Find a winning first move.
In the last-move-loses version, numbers \(1\pmod5\) are losing.
If one stone remains, the player to move loses. Thus the losing positions are \(1,6,11,\ldots\), that is \(1\pmod5\). Since \(47\equiv2\pmod5\), the first player takes \(1\) stone and leaves \(46\equiv1\pmod5\). Then the first player complements the opponent’s move to \(5\).
A stronger last-move-loses task.