Problem
In the game of Noughts, each player takes a turn to place a nought on the board (see below). Each new nought goes into a new square. The winner is the first person to place three noughts in a row. Is it possible for either the first player or the second player, to always win? (Assume that each player plays to win and plays as well as is possible.) If so, what is the winning strategy? If not, why not?

What is this problem about?
This problem develops the idea that games have strategies. That some games can be played well by a player but that player is still unable to win.

To get to this stage, the children will need to use logic as well as symmetry.

Achievement Objectives
Geometry (Level 4)
- apply the symmetries of regular polygons (in this case squares and rectangles)

Mathematical Processes
- devise and use problem solving strategies to explore situations mathematically (guess and check, make a drawing, use equipment)
- make conjectures in a mathematical context
- critically follow a chain of reasoning

Resources
counters
Blackline master of the problem (English)
Blackline master of the problem (Maaori)

Specific learning outcomes
The children will be able to:
- use symmetry as a game strategy
- apply problem solving strategies to a game context

Teaching sequence

1. Play a class game of Noughts. Let the class make the first move and you follow.
2. Play another game of Noughts with you starting.
3. Discuss their initial ideas about the game.
   What ideas do you have about Noughts after 2 games?
4. Pose the problem for the children to work on in pairs.
5. As the children play the game ask questions that focus them on describing their reasoning?
   What is a good (first) move? Why? How do you check that out?
6. Once the children have found the strategy check that they can identify the symmetries of the game strategy.
7. Give the children the opportunity to test out their strategy with another pair.
8. Share solutions – this may be in the following lesson giving the children the opportunity to test their strategies with other pairs and family members.

Extension
Try playing the game with a board with 11 squares. Will Player A still win? What if there were any odd number of squares on the board, 59, say. Who will win then? What happens if the board has an even number of squares?

Solution to the problem
The first player can always win if she plays correctly. Call the first player, Player A and the second player, Player B. To win, Player A needs to put her first nought in the centre square of the board. Then, wherever Player B goes, Player A should copy that move but on the opposite side of the board. We show this in the picture below.

Player A has put the first nought in the centre (we have shown it as A1 to indicate that it was Player A’s first move). Then Player B has put a nought in the square marked B1. Player A has replied by putting her next nought in square A2, to match Player B’s move.

Now suppose that Player B has a move that will not mean that Player A can get three noughts in a row. Then there must be a square on the symmetrically opposite side of the board that is safe for Player A. If Player B doesn’t have a safe move, then Player A wins and doesn’t play symmetrically.