# The Tell-Tale Board

## by Greg Egan

The applet below implements an ingenious puzzle. (I did not invent this puzzle myself. But I won’t link to any other source from this page because I don’t want to give spoilers.)

Take a chessboard, or chequerboard, and mark each square in some way with two alternatives, at random. You could place 64 identical coins heads or tails on each square, or you could place coins (or some other convenient marker) on some squares and not on others.

I claim that:

• Each configuration of the board encodes the location of some particular square on the board.
• It does this in a simple, systematic way that is easy to decode once you understand the principle; there is no extra obfuscation of the message that requires some kind of arbitrary key.
• There are some special configurations, S, with the following properties:
• You can always reach one of the configurations in S by changing the state of exactly one square.
• If you start with any configuration in S and change the state of exactly one square, the new configuration will encode the location of the square you changed.

Apart from the initial challenge of figuring out how the board’s configuration encodes the location of a square, this applet lets you practise performing the decoding yourself. If you and a friend become proficient at doing this, you can perform the following party trick:

In your absence, the guests mark all the squares of the board at random. Then your friend (openly) makes one change to the board, and invites the guests to make one other change.

You then enter the room, deduce what the second change was, and reverse it.

To use the applet, just click/tap on the square that you think is encoded by the current configuration. If you are right, the square will turn green; if you are wrong, it will turn red. If you have “Always show right answer?” selected, the square that was actually encoded will turn green.

Science Notes / The Tell-Tale Board / created Thursday, 2 January 2020 / revised Sunday, 5 January 2020