Olympiad Chessboard Problem

Question

I am stumped by this contest math training problem from South Africa:

Consider an 8 × 8 chessboard with the bottom left 3 × 3 squares occupied by cute little frogs. Each cute frog can jump over any frog adjacent to it, vertically or horizontally, to land on another square two squares away from where it started. Is it possible by repeating this operation to get all the cute frogs to the top right 3 × 3 squares?

I am wondering what kind of characteristic is associated with this setup. I tried parity and colouring, but to no success.

You can find a colouring argument – try to find invariants of how a frog moves. — Aphelli, Aug 28 '21 at 09:58

score 9 · Accepted Answer · edited Sep 05 '21 at 19:24

9

Color the columns black and white alternately.

Frogs from black squares can only move to other black squares. Now count the number of black squares in the bottom left $3\times 3$ and in the top right $3\times 3$

edited Sep 05 '21 at 19:24

nonuser

90,026

answered Aug 28 '21 at 10:00

acat3

11,897

1

Very nice! Notice that frogs can move also diagonaly and this still works. – nonuser Sep 05 '21 at 19:32
@nonuser But saying so divulges the solution of course. – Jan 12 '23 at 07:30

Olympiad Chessboard Problem

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