Mathematical Notes

0 Conversations

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.


Bookmark on your Personal Space


Conversations About This Entry

There are no Conversations for this Entry

Entry

A87862206

Infinite Improbability Drive

Infinite Improbability Drive

Read a random Edited Entry


Written and Edited by

Disclaimer

h2g2 is created by h2g2's users, who are members of the public. The views expressed are theirs and unless specifically stated are not those of the Not Panicking Ltd. Unlike Edited Entries, Entries have not been checked by an Editor. If you consider any Entry to be in breach of the site's House Rules, please register a complaint. For any other comments, please visit the Feedback page.

Write an Entry

"The Hitchhiker's Guide to the Galaxy is a wholly remarkable book. It has been compiled and recompiled many times and under many different editorships. It contains contributions from countless numbers of travellers and researchers."

Write an entry
Read more