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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View Full DocumentInductive Definitions
An inductive definition (also called a recursive definition) has two parts:
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 Spring '10
 fleck
 Logic, Mathematical logic, base case, Fibonacci number, IH F3k

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