MathDB

We are given a natural number

k

. Let us consider the following game on an infinite onedimensional board. At the start of the game, we distrubute

n

coins on the fields of the given board (one field can have multiple coins on itself). After that, we have two choices for the following moves:

(i)

We choose two nonempty fields next to each other, and we transfer all the coins from one of the fields to the other.

(ii)

We choose a field with at least

2

coins on it, and we transfer one coin from the chosen field to the

k-\mathrm{th}

field on the left , and one coin from the chosen field to the

k-\mathrm{th}

field on the right.

\mathbf{(a)}

n\leq k+1

, prove that we can play only finitely many moves.

\mathbf{(b)}

For which values of

k

we can choose a natural number

n

and distribute

n

coins on the given board such that we can play infinitely many moves.

Problem Statement