Competition problem — Combinatorics

Combinatorics Difficulty 5.0 AIME Prove it Brazil

Three numbered tiles are arranged in a tray as shown:

1	2
3

Show that we cannot interchange the

1

and the

3

by a sequence of moves where we slide a tile to the adjacent vacant space.

Solution

Write down the order of the tiles reading clockwise around the perimeter, starting at $1$ . We get $123$ and no move changes that, so we will always get $123$ after any sequence of moves. But the desired arrangement would give $132$ , so it is not possible.

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