# am_lect_19 - Another proof by strong induction Claim n N...

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Another proof by strong induction Claim : n N , Player 2 can win an ( n , n ) Nim game Proof by induction on n : Base case ( n = 1): Player 1 can only pick one stick, and Player 2 wins by picking the other stick Inductive step: Let k N such that m k , Player 2 can win a ( k , k ) Nim game [strong IH] We need to show that Player 2 can win a ( k +1, k +1) Nim game. Suppose Player 1 picks j sticks. Assume WLOG that this is from the first pile, so we are left with a ( k +1 j , k +1) game. There are two cases: Case 1: j = k +1, in which case Player 2 wins the (0 , k +1) game by picking j = k +1 sticks from the second pile Case 2: 1 j < k +1, in which case Player 2 picks j sticks and we are left with a ( k +1 j , k +1 j ) game, which Player 2 can win (by the strong IH). Hence the statement holds for k +1, which completes the proof by (strong) induction.

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Inductive Definitions An inductive definition (also called a recursive definition) has two parts:
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am_lect_19 - Another proof by strong induction Claim n N...

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