# Big Toe Noughts & Crosses

### Years 2 - 8

#### Summary

Noughts & Crosses, or Tic-Tac-Toe as it is sometimes known, is a game every child knows. The aim of the game is to be first with three counters in a row - horizontally, vertically or diagonally - to win. The attraction seems to be that you can sometimes win - of course you can sometimes loses to, but at other times you can draw and that is almost as good as winning. However when the question 'Is there a winning strategy?' is investigated is turns out that the game should also be a draw. That's very fair, but not very interesting. So, what happens if we add an extra place to the game and create Big Toe Noughts & Crosses?

#### Content

• logic and reasoning
• if-then reasoning
• concept of fairness
• psychology of game playing

#### Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

When analysing a game it must be assumed it is being played by the two best possible players. It's one thing to win a game by chance, or against a week player, but another thing entirely to win against the best possible opponent. With this approach it can be seen that standard Os & Xs should be a draw every time.

 The centre space is the most powerful because from it three in a row can be formed in 4 ways. Next is any corner space because that could form a three in 3 ways. The middle spaces are least powerful because they can only form a three in 2 ways.
So, using these best moves one example of play is...

 Player A plays in the centre. (Move 1) This move reduces the ways of making three in a row in a corner to 2 and in a centre to 1. So the best move now is in a corner. Player B plays in a corner. (Move 2) Player A plays in a corner not diagonally opposite Player B. (Move 3) Player B is forced by this move to play in a corner opposite Player A. (Move 4)

• Player A is forced to play in the middle between this last move and the other corner plug of Player B. (Move 5)
• Player B is forced to play opposite this move of Player A. (Move 6)
• Player A is still able to make three so takes Move 7.
• Player B is forced to make Move 8 and the only move left is Move 9 which Player A must make. Hence, a draw.
Any variation from this pattern, may lead to a win, but only because one or both players have not chosen the option with the most possible remaining ways of making three in a row. In fact, the aim of Os & Xs should be for both players to be so good they always force a draw. Something like a 0-0 draw in soccer, can still be a good, even great, game.

 Students soon realise that the extra space, the big toe, in Big Toe Os & Xs makes a different game. This diagram shows the number of ways of making three in a row for each space. Now Player A has a choice about where to start and having started their action changes the diagram of possibilities. Using Poly Plug Paper will help keep track of these changing possibilities as the game is analysed. Following the game through in a similar way to above leads to the conclusion that Player A must win.

So, this game is not fair. But it does produce a win. Is it an improvement on Os & Xs?

#### Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.

If you are using Menu Maths Packs one between two in your class, you have sufficient material to use this menu item as a whole class investigation.

Today we are all going to choose Big Toe Os & Xs from the menu.
The purpose in doing so would be to model Working Mathematically. Structure the lesson in a similar way to the outline above, and encourage students to record and perhaps publish their analysis. PowerPoint, or a sequence of photos, might be useful publishing tools.

#### Extension

• What happens if the game is Long Os & Xs?

#### Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.

Big Toe Os & Xs is not in any Maths With Attitude kit, but it is part of Poly Plug Game Pack resource book. This publication focuses on content-free reasoning based on four families of games, each of which has four variations. The Noughts & Crosses family is one of those families.