Prove that if and are sets such that , then
it is true for n=1.
Let P(k) be true.
We need to prove that P(k+1) is true.
It is true for P(k+1) is true.
Consider this variation of the game of Nim. The game begins with n matches. Two players take turns removing matches, one, two, or three at a time. The player removing the last match loses. Using strong induction to show that if each player plays the best strategy possible, the first player wins if or for some nonnegative integer j and the second player wins in the remaining case when for some nonnegative integer j.
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