Problem
COM-B1-M08-P019 Do Not Say \(64\)
#19
★★★☆☆ Level 3 of 5
Players alternately add a number from \(1\) to \(5\) to a total. The initial total is \(0\). A player whose move makes the total at least \(64\) loses. Who wins?
Positions \(63,57,51,\ldots\) are losing.
If the total is \(63\), any move makes the total at least \(64\), so the player to move loses. Thus \(63\) is losing. Earlier losing positions differ by \(6\): \(57,51,\ldots,3\). From any such position every move goes between neighboring losing positions, and from any other position one can move to the nearest losing one. From \(0\), the first player adds \(3\) and leaves a losing position. Then the first player complements the opponent’s moves to \(6\).
Good practice in backward analysis from a forbidden boundary.