MathDB

Consider an arrangement of tokens in the plane, not necessarily at distinct points. We are allowed to apply a sequence of moves of the following kind: select a pair of tokens at points

A

and

B

and move both of them to the midpoint of

A

and

B

.
We say that an arrangement of

n

tokens is collapsible if it is possible to end up with all

n

tokens at the same point after a finite number of moves. Prove that every arrangement of

n

tokens is collapsible if and only if

n

is a power of

2

Problem Statement