## Puzzle: Magic Square

October 29, 2013 § 7 Comments

Hello Math people. I heard this puzzle a few weeks ago, and I thought it was pretty great. Here is how it goes:

You and a friend are playing a game against an adversary. The game is played as follows. You walk into a room with the adversary, leaving your friend outside. The room contains a checker board (the typical 8 by 8 kind), and each square has on it exactly one coin. Each coin can either be heads-up or tails-up. Then, the adversary chooses exactly one square which he calls the magic square. You, then, based on the arrangement of the coins and which square is the magic square, will choose exactly one square and turn its coin over (either from heads-up to tails-up or from tails-up to heads-up). Then, you will leave the room through a back door and wait in a separate room. Your partner then comes into the adversary’s room, looks at the board, and proceeds to take a guess at which square is the magic square. If your partner guesses correctly, you win.

Assuming you and your partner have no idea of what arrangement the board will be in when you enter the room, come up with a strategy for you and your partner to use so that you can always win.

Note: the only information you or your partner can have from looking at a coin is which square it is on and whether or not it is heads-up. The solution does not involve doing something silly by rotating the coins; the adversary chooses the orientation of the coin after you flip it.

Thanks for reading, and feel free to message me if you would like the answer!

-A Student of Logic

Are you allowed to write on the coins? You could just write the chess coordinate on a coin.

No, unfortunately not. The idea behind the puzzle is simply that you are given the board covered in coins, you can turn over exactly one coin, and then your partner receives the board exactly as you received except the coin you chose is flipped. You could treat it as though you can not touch anything — you select the square and some machine flips the coin for you.

In that case, I don’t see a way that this is possible. There is no way for your partner to even tell which one is flipped or know what square your adversary will choose.

If the coins were placed neatly on the checkerboard with all the heads and tails in portrait position, you could place the head or tail so that it is the only one that appears to be upside-down. Otherwise, if the coins are all placed in the middle of each square, you could place your coin so that it is a little off-center and touches the side of its square.

As I tried to convey, the idea is not to do some spatial trick with the coins. The only information your partner can use is which squares have heads on them and which have tails. Assume the adversary chooses how each coin is placed, and you tell the adversary which coin you want to flip. Then, the adversary flips the coin for you.

Okay, I’ll keep thinking about it before I give up and ask for the answer.

I appreciate your optimism! Do not get discouraged — it is a very difficult problem.