Problem
COM-B1-M08-P022 From \(1\) to \(7\)
#22
★★★★☆ Level 4 of 5
There are \(2026\) stones. In one move, a player may take from \(1\) to \(7\) stones. The last move wins. Who wins, and what should the first move be?
Multiples of \(8\) are losing.
For moves from \(1\) to \(7\), the losing positions are multiples of \(8\). Since \(2026\equiv2\pmod8\), the first player takes \(2\) stones and leaves \(2024\), a multiple of \(8\). After that, the first player answers a move of \(a\) by taking \(8-a\), always returning a multiple of \(8\) to the opponent.
A large number should not hide the simple structure.