In 1935, T. R. Dawson, the prolific composer of fairy chess problems,
invented a little game now known as Dawson's Chess, involving only
pawns on a 3xN chessboard. He invested considerable effort trying
to find a perfect winning strategy, but was ultimately unsuccessful.

We now know that much of Dawson's difficulty arose from the winning
condition he imposed on the game: whoever makes the last move loses.
This is known as the misere play condition. The normal-play
counterpart of Dawson's Chess---in which the player who makes the
last move wins---was solved in 1956, but the original misere-play
formulation remains an open problem, after more than seventy years.

This is no accident: games with the misere play condition tend to be
vastly more difficult than games with the normal play condition (even
when the rules are otherwise identical). As a result, many authors
have dismissed the misere theory as essentially intractable.

However, a recent breakthrough, due to Thane Plambeck, has
sparked renewed interest. Plambeck showed that much of the
misere-play theory for a specific game G can be localized to a certain
commutative monoid, the misere quotient of G. I will describe
Plambeck's construction, and show how the misere quotient contains
exactly enough information to recover a perfect winning strategy for G.
Then I'll present misere quotients for several previously unsolved
games and discuss what little we know about the general structure
theory of misere quotients. Finally, I'll (hopefully) shed some light on
just why no one has yet solved Dawson's Chess.