Cubes Cubed

No 5 in a series of posts based on Thinking Mathematically (1985) by Mason, Burton, Stacey

I might use this task as an introduction before doing anything involving 3D visualisation. I’ve always felt that being able to conceptualise physical shapes is a skill that is very hard to teach in an instructive way. Tasks like this provide experiences that help to develop those skills. Here is the task:

I have 8 cubes all the same size. Two of them are painted red, two green, two blue and two yellow.
I wish to assemble them into one large cube with each colour appearing on each face.
In how many different ways can I assemble the cube?

The first challenge of this task is to interpret what it is asking and construct a mental image of that. The above image may help.

Then the question comes what do we mean by “different” large cubes, which is a conversation starter in itself. If we have the physical cubes in the form of multilink cubes we are likely to see that “different” means that one large cube cannot be simply rotated to form another one. The position of the small cubes actually have to be changed.

This 3D animation created on Geogebra (file here) may also help if real cubes are not available.

Feb-13-2017 09-26-28.gif

As John Mason points out in the book, whichever representation is used, it is important to record carefully the “different” cubes so they can be compared.

I came to the realisation that I need only look at the 4 small cubes on the front face because I can deduce where other 4 cubes on the rear face will be. Because no two small cubes of the same colour can share a face, the small cubes must be arranged so that they on diagonally opposite corners of the large cube.

screen-shot-2017-02-13-at-10-09-23So then it was a case of working out how many ways there are of arranging 4 things in a 2×2 arrangement where rotation is not allowed. I actually looked at it like this.

I convinced myself there are six ways of doing this.  But I’m still not convinced that this means 6 is the answer to the question.  I feel I would need to construct it to be sure I haven’t got any repeats within these 6.

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