A cube is about to get fully painted using $3$ different colors. Each color is being used for $2$ faces of a cube. How many different cubes can be created this way?

I saw this in a fifth grade math contest and it does not appear to be an easy problem. It took me almost ten minutes to figure out the answer (which happens to be $6$) but I did it by imagination and visualization which is not nearly as satisfactory as a rigid proof.

Could somebody offer an interpretation in terms of advanced mathematics? Group theory (see Burnside's lemma) or maybe graph theory is what comes to mind...

  • $\begingroup$ This seems to relate to Polya's Coloring Theorem: en.wikipedia.org/wiki/P%C3%B3lya_enumeration_theorem $\endgroup$ – Vladhagen Nov 11 '13 at 19:01
  • $\begingroup$ I don´t think you need advanced mathematics to solve this one. How many ways can I choose 2 objects out of 6? That´s after applying the first color. To that I multiply the number of ways to choose 2 objects out of 4 and I am left with two uncoloured faces which I will colour with the third colour. $\endgroup$ – Adam Nov 11 '13 at 19:23
  • $\begingroup$ You are missing (some of) the point, Adam. The cube has a geometric structure. That makes certain choices different from others. $\endgroup$ – Leen Droogendijk Nov 11 '13 at 19:36
  • $\begingroup$ Except, that if you rotate the cube you get, according to my calculation, a different colouring and yet it is clear that it is the same cube. So you need to divide my answer by something - number of rotations? $\endgroup$ – Adam Nov 11 '13 at 19:37
  • $\begingroup$ Yaeh, I am starting to realize it. $\endgroup$ – Adam Nov 11 '13 at 19:38

This one can be done with the Polya Enumeration Theorem, which requires the cycle index $Z(G)$ of the face permutation group $G$ of the cube. We now enumerate the permutations from this group by their cycle structure.

There is the identity, which contributes $$a_1^6.$$ There are rotations about diagonals connecting opposite vertices, which contribute $$4 \times 2 \times a_3^2.$$ There are the rotations about an axis passing through the centers of opposite faces, which contribute $$3\times (2 a_1^2 a_4 + a_1^2 a_2^2).$$ Finally there are rotations about an axis passing through the midpoints of opposite edges, which contribute $$6\times a_2^3.$$ It follows that the cycle index is $$Z(G) = \frac{1}{24} a_1^6 + \frac{1}{3} a_3^2 + \frac{1}{4} a_1^2 a_4 + \frac{1}{8} a_1^2 a_2^2 + \frac{1}{4} a_2^3.$$ Substituting the three colors red, green, and blue into this cycle index we get $$Z(G)(R+G+B) = 1/24\, \left( R+G+B \right) ^{6}+1/4\, \left( R+G+B \right) ^{2} \left( {R}^{4} +{G}^{4}+{B}^{4} \right)\\ +1/8\, \left( R+G+B \right) ^{2} \left( {R}^{2}+{G}^{2 }+{B}^{2} \right) ^{2}+1/3\, \left( {R}^{3}+{G}^{3}+{B}^{3} \right) ^{2}+1/4\, \left( {R}^{2}+{G}^{2}+{B}^{2} \right) ^{3}.$$ Expanding this cycle index we obtain $${B}^{6}+{B}^{5}G+{B}^{5}R+2\,{B}^{4}{G}^{2}+2\,{B}^{4}GR+2\,{B}^{4}{R}^{2}+2\,{ B}^{3}{G}^{3}+3\,{B}^{3}{G}^{2}R+3\,{B}^{3}G{R}^{2}\\+2\,{B}^{3}{R}^{3}+2\,{B}^{2 }{G}^{4}+3\,{B}^{2}{G}^{3}R+6\,{B}^{2}{G}^{2}{R}^{2}+3\,{B}^{2}G{R}^{3}+2\,{B}^ {2}{R}^{4}\\+B{G}^{5}+2\,B{G}^{4}R+3\,B{G}^{3}{R}^{2}+3\,B{G}^{2}{R}^{3}+2\,BG{R} ^{4}+B{R}^{5}+{G}^{6}+{G}^{5}R\\+2\,{G}^{4}{R}^{2}+2\,{G}^{3}{R}^{3}+2\,{G}^{2}{R }^{4}+G{R}^{5}+{R}^{6},$$ which finally gives $$[R^2G^2B^2] Z(R+G+B) = 6.$$ This generating function includes all distributions of three or fewer colors,e.g. the coefficient of $GR^5$ is one because there is only one way to paint the cube using one color five times and a second one for the remaining face. Note that we had rotations about face pairs, edge pairs and vertex pairs, which is a standard feature in the symmetry groups of regular polyhedra. The reader is invited to compute the cycle index of the symmetry group of the faces of a tetrahedron.

Here is another interesting calculation involving a cycle index.


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