MathDB

Let

(a,b)

be a pair of natural numbers. Henning and Paul play the following game. At the beginning there are two piles of

a

and

b

coins respectively. We say that

(a,b)

is the starting position of the game. Henning and Paul play with the following rules:

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They take turns alternatively where Henning begins.

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In every step each player either takes a positive integer number of coins from one of the two piles or takes same natural number of coins from both piles.

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The player how take the last coin wins. Let

A

be the set of all positive integers like

a

for which there exists a positive integer

b<a

such that Paul has a wining strategy for the starting position

(a,b)

. Order the elements of

A

to construct a sequence

a_1<a_2<a_3<\dots

(a)

Prove that

A

has infinity many elements.

(b)

Prove that the sequence defined by

m_k:=a_{k+1}-a_{k}

will never become periodic. (This means the sequence

m_{k_0+k}

will not be periodic for any choice of

k_0

)

Problem Statement