My friend likes to impress people by playing 3-5-7 which has three piles of counters of sizes 3, 5 and 7. You can remove any number of coins from a single pile, the last player to move loses.
This is a winning position for the first player, but With a solid understanding of the game tree she wins nearly every time playing second. She says, it reduces to knowing a few winning positions.
Two piles of the same size is second player win, in the jargon of Combinatorial Game Theory. Here is the pile (5,5).
If first player moves to (5,n) for n > 1, second player can imitate on the other pile, moving to (n,n). However, if first player moves to (5,1), second player moves to 1 and wins.
The other winning positions she remembers is (3,4,1) and (4,5,1). She can win once she recognizes these positions. Eventually (after losing many times) I told her that (n, n+1,1) is a losing position for any n...
Recently there was a theory of Misere quotients where each game has a commutative monoid assoicated with it. What does that monoid looks like here? Is it finitely generated?