No Three in a Line Game

The Problem

Mary the milk lady has a square milk crate that can hold 9 bottles. She plays this game with Fred. First she puts in a bottle and then he does. They keep alternating in this way. However, the rule is that no three bottles can be in a line. The winner is the one who puts the last bottle in the crate. If Mary always goes first, who should always win, Mary or Fred?

Teaching sequence

1. Talk about milk crates and their symmetry.
2. Tell the class Mary’s problem. Remind them of other games that they have considered as well as previous strategies for the no-three-in-line problems.
3. After some discussion, let the class play a series of game to see if they can find a pattern. Is there a winning strategy for either player?
4. Help the children that need it.
5. Call them all together from time to time to see what strategies they have come up with. Let the more able children go on to the Extension problem.
6. Try to get them to see the systematic approach that we use in the Solution.
7. Let a few groups report back to the whole class. They can illustrate what they are trying to say by playing a game.
8. Give the class the opportunity to write up their solutions to the problem.

Solution

Now Fred knows that the only way that he can win is if he forces the game to stop after 2, 4, or 6 bottles have been added to the crate. But from No-Three-In-A-Line Again, he knows that it has to be 4 or 6. So he is going to try to force the bottles into one of the positions below, or a position that is symmetrical with one of these positions.

On the other hand, Mary will win if she can get the game to go to an odd number – 3 or 5. But she knows that 3 isn’t possible. So she has to look for one of the 5 bottle positions below.

This game can now be analysed using the fact that a 3 by 3 crate essentially has only three different kinds of squares: (i) a corner square; (ii) a middle square on a side; and (iii) the centre square. This means that Mary only has three starting moves. But the centre square move is powerful. Although they both only have one winning end-game

with a bottle in the centre square, Mary is able to force the game to go her way. Watch.

Mary plays the centre square. Now, by symmetry, Fred has only two moves that he can make. In Game 1 above he plays in the middle square on an edge. In Game 2 he plays in the corner square.

Now to win, Mary only has to put in a bottle so that Fred’s winning position can’t be achieved. In both cases she plays in the corner as shown. Crosses show where Fred can’t play. Now wherever Fred plays, Mary can finish the game by completing the crate as in M above. Hence Mary can always win.