  A single-player game of Memory is played with $n$ distinct pairs of cards,
with the cards in each pair bearing identical pictures. The cards are laid
face-down. A move consists of revealing two cards, chosen adaptively. If these
cards match, i.e., they bear the same picture, they are removed from play;
otherwise, they are turned back to face down. The object of the game is to
clear all cards while minimizing the number of moves. Past works have
thoroughly studied the expected number of moves required, assuming optimal play
by a player has that has perfect memory. In this work, we study the Memory game
in a space-bounded setting.
  We prove two time-space tradeoff lower bounds on algorithms (strategies for
the player) that clear all cards in $T$ moves while using at most $S$ bits of
memory. First, in a simple model where the pictures on the cards may only be
compared for equality, we prove that $ST = \Omega(n^2 \log n)$. This is tight:
it is easy to achieve $ST = O(n^2 \log n)$ essentially everywhere on this
tradeoff curve. Second, in a more general model that allows arbitrary
computations, we prove that $ST^2 = \Omega(n^3)$. We prove this latter tradeoff
by modeling strategies as branching programs and extending a classic counting
argument of Borodin and Cook with a novel probabilistic argument. We conjecture
that the stronger tradeoff $ST = \widetilde{\Omega}(n^2)$ in fact holds even in
this general model.
