Try this: Magic noughts and crosses

You will need:

• Pen
• 2 pieces of paper
• A friend

What you do:

In order to do this trick, both you and your friend will need to know how to play noughts and crosses (or tic-tac-toe). If you don’t know how to play, or you want a refresher on the rules, click here

1. On a sheet of paper, write out a copy of the following board:
   O X O
   O X X
   X O X
   
   (If you want to remember this so you can do this trick anywhere, read the lines across the board: oxo, oxx, xox)
2. Fold the piece of paper so your friend cannot read it, and put it somewhere obvious, so you can reveal it later
3. Challenge your friend to a game of noughts and crosses. When they accept, draw the board, and put your X in the middle.
4. If they play on a corner, play on the square next to them that is clockwise around the board.
   If they play on a side, (not a corner) play on the square next to them that is anticlockwise around the board.
5. If they play sensibly, then the game will end in a draw. When you do, reveal the piece of paper that you have written on. If you spin the paper around to the right angle, the two games will be exactly the same!

What’s happening:

This trick is surprising because there are so many different moves that are made in a game. There are two main reasons why this trick works:

Firstly, your opponent doesn’t get much choice. On their second and fourth moves, your opponent has to play to block you from winning. This means that they only get two real choices.

Secondly, the board can be rotated at the end to match your prediction. This means that your prediction is not one, but four different predictions.

There are actually only eight possible results for a draw when the first player plays in the middle. Four of them are rotations of your prediction, and the other four are those answers, reflected in a mirror. If you follow the method above, you make sure to avoid the four answers that are a reflection of your prediction, so you will always get one of the rotations instead.

Applications:

Noughts and crosses is a very simple game, and with some practice, you can work out a strategy that means you will never lose. However, if you make a game just a little bit more complicated, it can be very difficult to work out the best strategy.

For example, you can play noughts and crosses in a cube instead of a square. This game is called 3D noughts and crosses, and is a lot harder than regular noughts and crosses. Because there are so many new ways to get a line, this game is usually played on a board four boxes to a side, and the winner must get four in a row. In 1980, Oran Patashnik proved that the first player in 4x4x4 3D noughts and crosses can force a win if they play perfectly.

Draughts is also a deceptively simple game. The rules can be written out on a single sheet of paper, and yet it took until 2007 for scientists to prove that if both players play perfectly the result is a draw. Go, an ancient Chinese boardgame, is even harder, even though the game has only ten rules.

Further information:

A video explaining the trick

3D noughts and crosses

tic tac toe on Wikipedia (more advanced)

Figure 1: Write out this board and keep it safe (note, this picture is rotated from the one in the instructions, but it's the same board)

Figure 2: If they play in the corner (shown in red), you play next to them, clockwise

Figure 3: If they play on a side (shown in red), you play next to them, anticlockwise