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All in the Mind

Imagine you are suspending a cube from one vertex (corner) and allowing it to hang freely. Now imagine you are lowering it into water until it is exactly half submerged. What shape does the surface of the water make around the cube?

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Painting Cubes

Imagine you have six different colours of paint. You paint a cube using a different colour for each of the six faces. How many different cubes can be painted using the same set of six colours?

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Tic Tac Toe

In the game of Noughts and Crosses there are 8 distinct winning lines. How many distinct winning lines are there in a game played on a 3 by 3 by 3 board, with 27 cells?

Noughts and Crosses

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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.

The interactivity can be used to give feedback to students.

Follow-up: How about trying to play Noughts and Crosses without paper?