Painting a cube using N different colors?

Question

In how many different ways can a cube be painted by using N different colors of paint?

Note that this question is not same to Painting the faces of a cube with distinct colours as the colours here may not be distinct (i.e. used many times). For Painting the faces of a cube with distinct colours, we must have $N>=6$, and the result will be:

$$\binom{n}{6}(5)(3!)$$

Can anyone help how to tackle such problem?

Isn't the title what you want to ask? You are asking the case $N=3$, right? — mathlove, Jan 20 '14 at 06:57
@mathlove Fixed the problem. I want to know is there a generalization solution for any N. Thanks. — herohuyongtao, Jan 20 '14 at 06:58

score 1 · Accepted Answer · answered Jan 20 '14 at 07:00

1

See here: http://en.wikipedia.org/wiki/P%C3%B3lya_enumeration_theorem

Specifically, under Examples -> Colored cubes.

answered Jan 20 '14 at 07:00

heropup

143,828

score 0 · Answer 2 · edited Jul 28 '23 at 14:20

0

I'm not sure if the answer is correct, but here's some of my idea: if we are painting a cube with $6$ distinct colours. fixing one of the faces, the face on the opposite side has $5$ colour choices using the circular arrangement for the rest of the $4$ sides we have $3!$ ways. In total, we have $5\times6=30$ ways.

Now if we have $n$ colours, after fixing the first face the opposite face has $(n-1)$ colour choices. There are now $(n-2)$ colours left to choose and for the four faces left we might choose $4$ colours from the $(n-2)$ colours and do the circular arrangements again.

In total, we have $(n-1)\times\binom{n-2}{4}\times3!$ ways.

edited Jul 28 '23 at 14:20

user1176409

1

answered Nov 07 '20 at 10:06

Alvey Lin

1

Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. – José Carlos Santos Nov 07 '20 at 10:26

Painting a cube using N different colors?

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