Problem
COM-B1-M08-P014 Moves \(1,2,4\)
#14
★★★☆☆ Level 3 of 5
There are \(30\) stones. In one move, a player may take \(1\), \(2\), or \(4\) stones. The last move wins. Who wins?
Try to prove that multiples of \(3\) are losing.
Multiples of \(3\) are losing: from such a position, moves \(1,2,4\) lead to residues \(2,1,2\) modulo \(3\), while from any non-multiple of \(3\) one can take \(1\) or \(2\) to leave a multiple of \(3\). Since \(30\) is divisible by \(3\), the first player loses with perfect play.
Shows that allowed moves need not be consecutive.