Painting an octahedron

Probability Level 4

You have 8 different colors, and you wish to paint the 8 faces of a regular octahedron, each in a different color. How many different ways can you do it?

Note : Different implies that for two coloring combinations $A$ and $B$ , you can't pick up $A$ and reorient it to make it look exactly like $B$ .

Image Credit: http://paulscottinfo.ipage.com/

The answer is 1680.

1 solution

Geoff Pilling
Sep 15, 2016

Solution inspired by @Seth Christman

First you can consider how many different ways there are to paint the $8$ faces. There are $8! = 40320$ .

Then we can count the redundancies, which is how many different ways you can orient the octahedron. First pick a base ( $8$ ways). Then rotate it ( $3$ ways). So there are $8*3 = 24$ different ways to orient the octahedron.

So, the total number of ways you can paint the octahedron is $\frac{8!}{24} = \boxed{1680}$ .

Painting an octahedron

Image Credit: http://paulscottinfo.ipage.com/

The answer is 1680.

1 solution

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