  We play a variation of Nim on stacks of tokens. Take your favorite increasing
sequence of positive integers and color the tokens according to the following
rule. Each token on a level that corresponds to a number in the sequence is
colored red; if the level does not correspond to a number in the sequence,
color it green. Now play Nim on a arbitrary number of stacks with the extra
rule: if all top tokens are green, then you can make any move you like. On two
stacks, we give explicit characterizations for winning the normal play version
for some popular sequences, such as Beatty sequences and the evil numbers
corresponding to the 0s in the famous Thue-Morse sequence. We also propose a
more general solution which depends only on which of the colors `dominates' the
sequence. Our construction resolves a problem posed by Fraenkel at the BIRS
2011 workshop in combinatorial games.
