# Colored Cubes

2008/08/04

Lets color the six faces of a cube. If we use six colors and each face has a different color, we get the known 30 MacMahon cubes. For this set a lot of problems were stated, analyzed and solved.
If we use more or less colors and allow two or more faces to have the same color, we get the following table.

Colors 1 2 3 4 5 6 7 8 n
Number of cubes Same colored faces allowed 1 10 57 240 800 2226 5390 11712 1/24*(n^6+3n^4+12n^3+8n^2)
All faces different - - - - - 30 210 840 1/24*n!/(n-6)!

According to the Polya-Burnside formula the number of cubes with same colors on their faces can be determined for general n.

What kind of problems can be solved with these sets? Perhaps you would like to make a box with the cubes of a set. No problem, but if the touching faces of the cubes must match in color, it's not so easy. Additionally a special coloring of the outside faces would be nice. I tried checkerboard coloring and uniform coloring of opposite faces. Click the number of cubes in the table to see some constructions.

Below you see a sliced box with outside faces in green, yellow and red where black, blue and white are hidden inside.

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