The national curriculum requires that pupils at all levels should be taught to reason. For example, at key stage 1, “pupils should be taught to explain their methods and reasoning when solving problems involving number and data”, and to “use mathematical communication and explanation skills”. By KS3 they “should be taught to show step-by-step reasoning in solving a problem”, to “explain and justify inferences and deductions using mathematical reasoning”, and to “recognise the importance of assumptions when deducing results”.

The Microsoft computer game Minesweeper can help to develop reasoning skills. The idea of the game is to identify the positions of a number of mines in a rectangular grid. Play starts by left-clicking the mouse in one of the cells of the grid. Three things can happen: if the cell contains a mine, it explodes and the game is lost; if the cell has no mine, a number appears indicating how many adjoining cells contain mines; if no adjoining cell contains a mine, all neighbouring cells are exposed, up to a boundary of cells that do adjoin mines.

Subsequent moves consist of left-clicking another cell (with the same possible results) or right-clicking a cell, which places a flag in it (used to mark a known mine). The best way to understand is to play a few games!

At beginner level, 10 mines are placed at random in an 8 by 8 grid. The position after a few moves is shown in figure 1: (white cells have been exposed, grey are unknown).

The cell marked A must contain a mine. This is because the cell to its south-west adjoins only one mine, and cell A is the only possibility.

This implies that all the cells marked B are free from mines, because they adjoin the cell north of A that touches only one mine (which is A). Finally, the three pairs of question marks indicate positions where one or other (but not both) of the pair must be a mine. (Explain!) Consequently, the cell at C must be free of mines, since the cell to its north-west is already touching a mine.

Children soon learn to recognise the situations occurring in the cells marked A and B, but the situation at C is more subtle. The game allows for a wide range of ability.

In a lesson, students need to play a few times to get used to the game. They can use reasoning on questions such as:

* Are there any cells that certainly contain a mine? How can you be sure?

* Are there any cells that certainly do not contain a mine? How can you be sure?

* Are there any positions where it will never be possible to be sure of the location of a mine?

After marking the mine at A and exposing the cells at B and C, we arrive at what is shown in figure 2. It is more difficult to make the next deduction, so some students will not see it unaided. However, many will be able to follow another student’s explanation.

The cell between cells E and F touches two mines, which must take up two of the three positions marked E, F and G. However, cells E and F cannot both contain mines because they share contact with a cell touching only one mine (south-west of F).

This leaves two possibilities, both involving a mine at G, which means there cannot be one at H. At present, it is impossible to decide between the two possibilities for cells D, E and F: there could be a mine at E only, or mines at both D and F. However, this is not a case that will remain ambiguous, because the status of F is likely to become clear after further moves.

Figure 3 shows the final position of the game. Figure 4 gives a genuine example of irresolvable ambiguity, where the last two mines could go in any of the four cells in the north-west corner.

A Minesweeper lesson can easily include interactive whole-class teaching, individualgroup work and an effective plenary.

* Two Mathematical Association publications also tackle the issue of reasoning: Can You Prove It? by Sue Waring (pound;10 non-members; pound;7 members) and Are You sure? Learning about Proof, by Doug French and Charlie Stripp (pound;8.50 non-members; pound;6 members). From MA Publications, 259 London Road, Leicester, LE2 3BE. Tel: 0116 221 0013.

Steve Abbott is immediate past president of the Mathematical Association

MICROSOFT MINESWEEPER

Minesweeper comes as part of the Microsoft Windows package. It is installed on most PCs (although not all networked PCs as the network manager may not want people “wasting” their time). If it isn’t installed you can click the START button and go to Settings, then Control Panel, then AddRemove programmes and click Windows set-up. Double click on Accessories from the list and click the box next to Games until a tick appears. Click OK on the next two screens and Windows installs the free games (Solitaire, Minesweeper, Freecell and Hearts). You may well need to insert the original Windows installation CD.