Counting Cube Colorings with the Cauchy-Frobenius Formula and Further Friday Fun Daniel Frohardt Wayne State University December 3, 2010 We have a large supply of squares of in 3 different colors and an ample supply of tape. The squares are all of the same size, and we can tape 6 of them together to form a cube. How many fundamentally different cubes can we make? Before setting out, we need to be precise about what we consider to be essentially the same coloring. For example, if we make the top face red and the other faces green, this is essentially the same as making the bottom face red and all other faces green. Let us agree that if one of the cubes can be rotated to look exactly like the other then they have essentially the same coloring. It s possible to enumerate all the ways to do this, using careful and systematic analysis, but there is a sophisticated way to do this using the group theory and the Cauchy-Frobenius Formula that generalizes to counting the number of colorings of other solids and planar figures. We can also count the number of essentially different ways to color the edges. 1 Squares To illustrate the principles involved, consider the simpler problem of counting the number of essentially different colorings of a square using two colors. If we do not allow any symmetries of the square, there are sixteen different ways to color the four sides of a square using two different colors (or possibly just one of them). Each side can have one of the two colors and 2 4 = 16. The following picture shows these 16 colorings. 1
In the following picture we label each square with the number of red edges. Since there are two fundamentally different ways to have two red edges, one with the red edges opposite each other, the other with them adjacent, we label them 2A and 2B. 2
0 1 1 2A 1 2B 2B 3 1 2B 2B 3 2A 3 3 4 1.1 Symmetries of the Square Let us enumerate the symmetries of the square. There are four rotations, by 0, 90, 180, and 270. The first of these is the identity map, which has order 1, and the 90 and 270 rotations have order 4. There are also four reflections, about the four axes of symmetry, one vertical, one horizontal, and two diagonal. Symmetry Order Name Identity 1 I 90 rotation 4 ρ 180 rotation 2 ρ 2 270 rotation 4 ρ 3 Reflection in NS-axis 2 σ Reflection in EW-axis 2 ρ 2 σ Reflection in NW/SE-axis 2 ρσ Reflection in NE/SW-axis 2 ρ 3 σ Compare the number of different manifestations of each type with the number of symmetries each example admits. Basic fact: The number of times each type occurs is 8 divided by the number of symmetries exhibited by a sample. 3
2 Counting The key idea in the proof is to count the number of elements in a carefully chosen set in two different ways. If we let P be the set of possible colorings and G be the group of all symmetries, we define X to be {(p, g) : p P, g G, and g(p) = p}. The two ways to count X are (1) to choose p first and count the number of group elements that fix p and (2) to run through the group elements first and count the number of patterns each group element fixes. The stabilizer G p of a pattern p is defined to be the set of all group elements fixing p, so X = G p. p P The previous analysis shows that G p = G / the number of essentially similar patterns. If we group the patterns by ones that are essentially similar we see that each set of similar patterns contributes G to the sum so X = r G where r is the number of essentially distinct patterns, which is what we re trying to compute. Counting the other way, X = Fix(g) where Fix(g) is the number of g G patterns fixed by g. Fix(g) is actually fairly easy to count. It is simply n c where n is the number of colors and c is the number of orbits of g on the individual components (edges or faces in the examples we have looked at). 3 The naïve approach It s possible to answer this question by a careful, but relatively unsophisticated count. 3.1 One color There are three ways to color the cube using just one of the three colors. 3.2 Two colors With two colors, if we make one face one color and the other five a different color it doesn t matter which face we single out since we can always rotate it to be the top one. So there are six ways to color a cube with one face one color and the other five faces a different color: we have three choices for the first color and two for the second. If we continue to use only two colors, but use one color for two faces and the other for four, then there are essentially two ways to do this after we have chosen our dominant and secondary color. The two specially colored faces could be opposite each other or adjacent. This means that we can do four of one color and two of another in twelve different ways. With two colors evenly used, that is, for three faces each, the two colors are not distinguishable by dominance, so there are only three ways to choose them. 4
Simply pick the color not used. Having picked the colors, there are two patterns. We could paint all of the faces adjacent to one corner one color and all of the faces adjacent to the corner opposite it the other color, or we could paint two opposite faces, plus one face sharing an edge with each of them, one color and the remaining three faces which must also consist of two opposite faces and one with a joint edge with each the other color. [It may be a good idea to think about this to convince yourself that these are the only two possibilities.] This shows that there are six possible ways to paint three faces one color and three faces another color. So far we have seen that there are three ways to do the coloring with a single color and twenty-four ways to do it with two of the three colors. 3.3 Three colors If we use all three colors, then we can have four faces one color, and the other two of differing colors; three of one, two of a second, and one of the third; or two of each color. 3.3.1 Four one one If we use two colors for only one face each, then either those two faces are adjacent or they are opposite. A bit of thought shows that there is essentially only one way to do each for any given choice of colors. There are three choices for the dominant color, and two ways to pick the two special faces, making six ways to color the cube this way. 3.3.2 Three two one Suppose we color three faces red, two blue, and one yellow. How many ways can we do this? If the three blue faces are all adjacent to the same corner of the cube, then there is just one way to do this. The two red faces must be adjacent, and also adjacent to the yellow, so we can rotate the cube so that the yellow face is on top and the red faces are in front and on the left. On the other hand, if the red faces are in the pattern such as left, front, right, then the two red faces can be either opposite or adjacent. If they are opposite then they have to be top and bottom. If they are adjacent, then one of them must be in back, and we can rotate the cube so that the yellow one is on top. This shows that for each choice of one color to use three times, one to use two times, and one to use once, there are three ways to color the cube. Since there are six ways to order the three colors, we have eighteen different ways to color the cube with this unequal apportionment of three colors. 3.3.3 Two two two Finally, we come to the most complicated case, where we use the three colors for two faces each. We break this down into subcases, depending on how many pairs of opposite faces get colored the same color. 5
If all three do, then we can put the blue in front and the red on the left, and the colors of the remaining faces will be forced: top and bottom yellow, back blue, and right red. There is just one way to do this. Suppose only one pair of opposite faces matches. We may as well assume that the front and the back are both blue. We can rotate the cube so that the top and left are both red. Once we pick blue as the color of the matching opposite faces there is just one way to do this. Since we could have made the matching pair one of the other colors, there are three ways to do this. This leaves us with the case where each of the three colors is used for two adjacent faces. We can rotate the cube so that the blue faces are front and top. Either the left or the right face (but not both!) must be red. So rotate to make the red face on the left. Then the right face is yellow. It requires a bit of thought, but if we do not allow mirror images to count as the same, then there are two essentially different ways to finish the coloring: either make the back face yellow and the bottom face red or vice versa. Adding everything up, there are fifty-seven ways to color the cube if we regard cubes that can be rotated to look like each other as the same. 4 Power tools Is there a better way to do this? Yes, there is. 4.1 Symmetries of the cube We observe that there are exactly 24 rotations of the cube. We can put any one of the 6 faces in front, and after doing so we can put any of the 4 adjacent faces on top. We can catalogue these rotations by considering the axes of symmetry. What are the axes of symmetry of a cube? There are three axes that go from the center of one face to the center of the opposite face. We can rotate the cube a quarter turn, a half turn or three quarters of a turn about any of these. Note that rotating three quarters of a turn about an axis, viewed from one end, is the same as rotating one quarter of a turn about the axis viewed from the other. There are thus six rotations by 90 and three rotations by 180 about these axes that join the centers of opposite faces. So far we have nine rotations. What are the others? Another axis of rotation joins opposite corners. There are four such axes, and we have two different rotations of one third of a turn about each. There are eight such rotations in all. We now have seventeen rotations, so we are missing seven. The do-nothing rotation is in fact a symmetry of the cube, so this means that we only need six more. 6
We need some more axes of rotation. We have joined face centers to face centers and corners to corners. What we haven t looked at yet are the axes that join the centers of opposite pairs of edges. The cube has twelve edges, so they come in six pairs. This is nice since we get six axes this way, for each of which we have a symmetry by making a half turn about it. To sum up: Symmetries of the cube Description Number Identity 1 One-half rotation, face axis 3 One-quarter rotation, face axis 6 One-third rotation, corner axis 8 One-half rotation, edge axis 6 The rotations of the cube form a group. We ll be using a couple of facts from group theory, but we do not need to go through the formal definition. Let us merely observe that the composition of two rotations is another rotation and this composition is associative. Furthermore, any rotation can be undone, so there is a cancellation law. If two rotations followed by the same final rotation produce the same result, then they must have been the same. 4.2 A useful formula Suppose we pick a coloring. How many other, superficially different, colorings are essentially the same in that they can be rotated into it? On general principles, this number is inversely proportional to the number of rotations that leave the coloring unchanged. This should seem intuitively plausible the more rotations we use up doing nothing, the fewer we ll have to move it to something different but we will look for a more convincing argument. The key observation is that the number of ways to move the coloring to another equivalent coloring is always the same no matter what we choose for the second coloring. Take, for example, the coloring where one face is red and the other five faces are green. Any of the four rotations about the axis through the red face will leave that coloring unchanged. Any of the other rotations will move it to a different one. There are a total of six superficially distinguishable colorings that are equivalent to this one the red face can go in any of six possible places. Note that 6 = 24/4. As another example, take the coloring where all three faces adjacent to the upper left front corner are red and the rest are green. There are just three rotations that leave this coloring alone, all of them rotations about the axis through that corner, and there are eight equivalent colorings, corresponding to the eight corners of the cube. 7
Note that 8 = 24/3. Similarly, there are three ways to color opposite faces red and everything else green, and for each such coloring there are eight ways to rotate the cube and leave the coloring alone: four rotations about the axis through the two faces and four rotations about the axes parallel to those faces. [The latter rotations come in two sub-types.] This time, 3 = 24/8. We need some notation. Let C be a coloring of the cube, that is, an arrangement of colors on the faces of the cube. Example: blue on top, green around the sides, and yellow on the bottom. Let Stab(C) be the set of all rotations C unchanged. Let n(c) be the number of superficially different colorings that are equivalent to C. We argue that n(c) = 24/ Stab(C). If r 1 and r 2 both rotate C to the same arrangement, then r 1 followed by r2 1 will belong to Stab(C). In other words, r 1 is the same as s followed by r 2 for some s in the stabilizer of C. r 1 and r 2 have the same effect on C if and only if r 2 = sr 1 for some s in Stab(C). Let C 1 = C, C 2, C 3,..., C k be the different colorings equivalent to C. Then the rotations of the cube can be divided into k sets, R 1, R 2,..., R k where every rotation in R i takes C to C i. Furthermore, every set R i contains exactly Stab(C) rotations. We can count the number of rotations of the cube in two ways: It s the number of ways to leave C alone, plus the number of ways to take C to C 2, plus..., plus the number of ways to take C to C k, which is just R 1 + R 2 + R 3 +... + R k, or k Stab(C). It is also known to be 24. So we have k Stab(C) = 24. Since k is just n(c) we have n(c) = 4.3 Counting a set in two ways 24 Stab(C) The technique of counting the same set in two different ways is a surprisingly potent one. We ll use a Nineteenth Century application of this technique. Let S be the set of all ordered pairs (C, r) where C is a coloring of the cube and r is a rotation of the cube that fixes C. To compute S we can pick C first, and add up the number of ways to pick symmetries that fix C. We can also, for each symmetry, can figure out how many colorings it fixes and add all those quantities together. Since this is a well-defined 8
number, if we count accurately, we ll get two different quantities that must be the same. Using the first approach, it is helpful to group the equivalent colorings together. The number of colorings equivalent to C is 24/ Stab(C). Each of them is fixed by Stab(C) rotations, so their total contribution to the number of pairs we were counting is just 2. This is true for every set of equivalent colorings. The total number of pairs we are counting is 24 times the number of types of equivalent colorings. So the number of types of equivalent colorings is 1/24 of the sum all Fix(S), for all types of rotation S. We next proceed to compute Fix(S) for each type of S. If S is the identity, it fixes everything, all 3 6 colorings. If S is a half rotation fixing a pair of oppisite faces then we can color those two faces with any of the 3 colors, and color each of the remaining opposite pairs with matching colors. We choose any of 3 colors 4 times, making 3 4 colorings. For quarter rotations, the two fixed faces can have any colors, but the remaining four must be the same. There are 3 3 colorings here. Similarly, for the corner rotations, we choose colors twice, once for the faces adjacent to a given corner, and once for the faces adjacent to the opposite corner. There are 3 2 colorings for each corner rotation. Finally, for the edge rotations, we have three pairs of colors to match up and 3 3 colorings for each such rotation. We summarize this. Description of S Number of such S Fix(S) Identity 1 3 6 One-half rotation, face axis 3 3 4 One-quarter rotation, face axis 6 3 3 One-third rotation, corner axis 8 3 2 One-half rotation, edge axis 6 3 3 This leads to the formula NOC = (3 6 + 3 3 4 + 6 3 3 + 8 3 2 + 6 3 3 )/24 NOC = (3 6 + 3 3 4 + 12 3 3 + 8 3 2 )/24 This is easy to evaluate. And we can replace 3 by any other positive integer. The general formula for n colors is: NOC = (n 6 + 3 n 4 + 12 n 3 + 8 n 2 )/24 Some results appear in the table below. 9
number of colors number of cube colorings 2 10 3 57 4 240 5 800 6 2226 7 5390 8 11712 9 23355 10 43450 11 76351 12 127920 13 205842 14 319970 15 482700 16 709376 17 1018725 18 1433322 19 1980085 20 2690800 10