MathDB

Consider a game played on the integers in the closed interval

[1, n]

. The game begins with some tokens placed in

[1, n]

. At each turn, tokens are added or removed from

[1, n]

using the following rule: For each integer

k \in [1, n]

, if exactly one of

k - 1

and

k + 1

has a token, place a token at

k

for the next turn, otherwise leave k blank for the next turn. We call a position static if no changes to the interval occur after one turn. For instance, the trivial position with no tokens is static because no tokens are added or removed after a turn (because there are no tokens). Find all non-trivial static positions.

Problem Statement