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 multilink 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 rightmiddletop 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 middlemiddlemiddle:
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(n – x + 1)
Diagonals: 6n(n – x + 1)
Long diagonals: 4(n – x + 1)
Total: (3n² + 6n + 4)( n – x + 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:
A marble is placed in the unit cube at leftmiddlebottom.
Another is placed at middlemiddlemiddle.
Where should the third marble be placed to make a winning line of three marbles?
How many winning lines go through middlemiddlemiddle?
How many are there altogether?
What strategies can you suggest to always win this game of 3D noughts and crosses?
