Solution
Chomp on the Divisors of 360
Show the problem again
The 24 positive divisors of 360 are written on a board. Two players alternate turns. On a turn the mover names a divisor still on the board and erases it together with all of its multiples still on the board. Whoever is forced to name 1 loses (naming 1 at any time loses). The first player can win, and exactly one first name wins. What is it? Answer with an integer.
Worked solution
Write each divisor as 2^a 3^b 5^c with a ≤ 3, b ≤ 2, c ≤ 1. Naming d erases the whole up-set of d, so this is three-dimensional Chomp on a 4×3×2 box, with 1 as the poisoned corner. Strategy stealing shows that the first player wins: if naming 360 (the top element, which erases only itself) lost, the winning reply of the second player would be a name that the first player could have used immediately, which is a contradiction. But the argument does not say which name wins. An exhaustive search over the reachable positions (the order ideals of the divisor lattice) shows that exactly one opening name wins, namely 6 = 2·3, and that all 22 other non-trivial names lose.
Source: Divisor form of Chomp due to Fred Schuh (1952). Statement written for AxiomIQ.