Noughts and crosses
The ideas in this problem have been expanded and made into another, more detailed problem. See Marbles in a Box.
Imagine a $3 \times3 \times3$ cube, made up from 27 unit cubes, all of which are made from clear plastic that can be filled with ease.
The location of a unit cube is described according to the following positions with respect to the three axes or directions:
- 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 different types of winning lines are there?
How many winning lines are there altogether?
How many winning lines of four are there altogether in a $4 \times 4 \times 4$ cube?
How many winning lines of $n$ are there altogether in an $n \times n \times n$ cube?
This problem will feature in Maths Trails - Visualising, one of the books in the Maths Trails series written by members of the NRICH Team and published by Cambridge University Press. Maths Trails - Visualising is due to be published later this year, but for more details about the other books in the series, please see our publications page .
As Allen (Sha Tin College) discovered, the trick to cracking this problem is to work systematically:
"My method was to start of with a small cube and work onwards from that. I started off with a $3 \times 3 \times 3$ cube and stared counting the number of lines for each Vertical, Horizontal and Diagonal"
Doing this he discovered that there are 13 winning lines that go through middle-middle-middle.
The winning lines in general seperate into three types:
"Lines" of three cubes can be made from cubes joined face to face
"Diagonals" are cubes joined edge to edge in a line - so the diagonal of a face for example
"Long Diagonals"are lines of three cubes joined vertex to vertex, going through the middle from a vertex to one diagonally opposite.
Counting these gives 27 lines, 18 diagonals and 4 long diagonals for the $3 \times 3 \times 3$ cube. In general, for an $n \times n \times n$ cube:
Lines: $3n^2$
Diagonals: $6n$
Long Diagonals: 4
This is a good opportunity to explore aspects of generality in three dimensions and for pupils to discuss images and possible solutions, and find convincing arguments for the unique solution.
For some this may be a hard problem to visualise, which is not always made easier by diagrams.
Suggestion for introducing the problem:
Invite the class to imagine a 3 x 3 square grid.
How many small squares are there?
How are they arranged?
Ask the group to draw a 3 x 3 grid.
Now, ask them, in their mind's eye, to colour one of their nine squares and ask for a volunteer to describe to the rest of the class where it is. When the person has described their square ask everyone to fill in that square on their grid. Repeat this activity several times with the aim of identifying some notation that fully describes the position of squares on the grid.
Repeat the activity but this time describe lines of three squares, like the winning lines in "Noughts and Crosses".
How many lines are there?
Suggestions for the main part of the lesson:
Ask the group to imagine that they have a 3 x 3 x 3 cube made up from 27 unit cubes.
Can they devise a notation for describing positions of little cubes and lines of cubes within this cube (as if they were playing 3D Noughts and Crosses)?
Tell the group that a marble is placed in the unit cube found at middle-middle-top. Another is placed at middle-middle-middle. Where should the third marble be placed to make a winning line of three marbles? Try some other examples with the group.
Ask students to come up with examples of their own.
Follow-up: How about trying to play Noughts and Crosses without paper?