MathDB

In each vertex of a regular

n

-gon

A_1A_2...A_n

there is a unique pawn. In each step it is allowed: 1. to move all pawns one step in the clockwise direction or 2. to swap the pawns at vertices

A_1

and

A_2

. Prove that by a finite series of such steps it is possible to swap the pawns at vertices: a)

A_i

and

A_{i+1}

for any

1 \leq i < n

while leaving all other pawns in their initial place b)

A_i

and

A_j

for any

1 \leq i < j \leq n

leaving all other pawns in their initial place.
Proposed by Matija Bucic

Problem Statement