Mathematical Notes
Created | Updated Feb 28, 2017
McMahon Cubes
These are cubes with each face painted a different colour. There are six possible colours. The full set of McMahon has every possible arrangement of colours with no two cubes being the same.
In calculating the number of McMahon cubes, we must eliminate cubes which are the same but rotated. So for example, if we number the six colours as 1 - 6 and call the faces of the cube front, back, left, up and down, then these two cubes are identical:
F=1, B=2, L=3, R=4, U=5, D=6
F=1, B=2, L=6, R=5, U=3, D=5
The second cube is the first one rotated 90° clockwise around the front face.
If we put the colour 1 at the bottom (which we can do by rotating the cube, then there are five different possible colours for the top face. If we now turn the cube so that the face with the largest number is at the back, then the other three colours can occupy the remaining three faces (front, left, right) in six different ways (ABC, ACB, BAC, BCA, CAB, CBA). This means that there are 30 possible patterns for the colours on the cubes, and a full set of McMahon cubes has 30 cubes.
What to do with the cubes?
McMahon set the problem of selecting one cube from the set, then building a 2x2 replica from 8 of the remaining cubes. The colours inside the cube must match - you can't put a red face touching a blue face for example.
"Instant Insanity" was a popular puzzle in the late 1960s / early 1970s. It had four cubes with faces in four different colours, so it wasn't McMahon cubes. You had to build a tower of four cubes so that no side had a repeated colour.