# Noughts and Crosses Teachers Notes

 Дата канвертавання 24.04.2016 Памер 13.63 Kb.

Enrichment in Mathematics – Algebra

#### Teachers Notes

This problem makes an excellent visualisation activity. It can be done as a whole class, in pairs or groups. Teachers may want to provide multi-link cubes and/or isometric paper, or layers of tracing paper. Geometry software could be used to demonstrate the solutions, or 3D models.
To extend the activity, pupils can investigate the number of winning lines:

• winning lines of four in a 4 x 4 x 4 cube

• winning lines of n in an n × n × n cube

• winning lines of 3 in an n × n × n cube

• winning lines of 3 in an n × n × n cube

• winning lines of 3 in a w × l × h cuboid

## Solution

The marble should be placed at right-middle-top to make a line of three.
The winning lines in general separate into three types:

Straight lines of three cubes made from cubes joined face to face.

Diagonals of cubes joined edge to edge in a line

Long diagonals of lines of three cubes joined vertex to vertex.
There are 13 winning lines that go through middle-middle-middle:

1 straight & 2 diagonal in each plane plus the 4 joining the vertices

There are 58 winning lines in total:

9 straight vertical and 9 straight in each horizontal plane = 27

2 diagonals in each vertical row = 6 vertical diagonals, + similarly 12 horizontal = 18

4 joining the vertices

Winning Strategy

take centre and a vertex to maximise the possible winning lines.

Generalising

n × n × n cube, lines of length n:

Lines: 3n²

Diagonals: 6n

Long diagonals: 4

Total: (3n² + 6n + 4)
n × n × n cube, lines of length 3:

Lines: 3n²(n – 2)

Diagonals: 6n(n – 2)

Long diagonals: 4(n – 2)

Total: (3n² + 6n + 4)(n – 2)
n × n × n cube, lines of length x:

Lines: 3n²(nx + 1)

Diagonals: 6n(nx + 1)

Long diagonals: 4(nx + 1)

Total: (3n² + 6n + 4)( nx + 1)

### Noughts and Crosses

Imagine a 3 x 3 x 3 cube, made up from 27 unit cubes.

Each cube can be filled with either a marble (O) or a cross (X).

The position of each unit cube is described using the three directions and:

• Left, middle, right

• Front, middle, back

• Top, middle, bottom

A marble is placed in the unit cube at left-middle-bottom.

Another is placed at middle-middle-middle.

Where should the third marble be placed to make a winning line of three marbles?

How many winning lines go through middle-middle-middle?

How many are there altogether?

What strategies can you suggest to always win this game of 3D noughts and crosses?

Worcestershire Numeracy Team Enrichment Activities

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