Painting Cubes
Problem
A large rose-tree stood near the entrance of the garden: the roses growing on it were white, but there were three gardeners at it, busily painting them red....
"Would you tell me please," said Alice, "why you are painting those roses?"
Five and Seven said nothing, but looked at Two. Two began in a low voice, "Why, the fact is, you see, Miss, this here ought to have been a red rose-tree, and we put a white one in by mistake; and, if the Queen was to find it out, we should all have our heads cut off, you know."
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Imagine you have some wooden cubes. You also have six paint tins each containing a different colour 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? Remember that two cubes are different only when it is not possible, by turning one, to make it correspond with the other. |
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Student Solutions
30 different painted cubes.
Let the six faces be painted a, b, c, d, e, and f.
With face a opposite face b there are six arrangements for the other four colours around the cube: cdef, cdfe, cedf, cefd, cfde and cfed.
Likewise for the face a opposite face c; face a opposite face d; face a opposite face e; and face a opposite face f. All have six arrangements for the remaining four colours.
Hence the total is 5 x 6 = 30 arrangements.