  In a recent paper Velleman and Warrington analyzed the expected values of
some of the parameters in a memory game, namely, the length of the game, the
waiting time for the first match, and the number of lucky moves. In this paper
we continue this direction of investigation and obtain the limiting
distributions of those parameters. More specifically, we prove that when
suitably normalized, these quantities converge in distribution to a normal,
Rayleigh, and Poisson random variable, respectively.
  We also make a connection between the memory game and one of the models of
preferential attachment graphs. In particular, as a by--product of our methods
we obtain simpler proofs (although without rate of convergence) of some of the
results of Pek\"oz, R\"ollin, and Ross on the joint limiting distributions of
the degrees of the first few vertices in preferential attachment graphs.
  For proving that the length of the game is asymptotically normal, our main
technical tool is a limit result for the joint distribution of the number of
balls in a multi--type generalized P\'olya urn model.
