Date: 15 June 1982 1045-EDT (Tuesday) From: Dan Hoey at CMU-10A To: Cube-Lovers at MIT-MC Subject: Patterns for Rubik's Revenge Message-Id: <15Jun82 104503 DH51@CMU-10A> CHECKERBOARDS I confirm Dave Plummer's result that it is impossible to make a checkerboard on all faces of a cube of even side. Proving this for the 2^3 is sufficient, and implies the corollary I sent last year that it is impossible to make S or Z or Zig-Zag patterns on all sides of the 3^3. This time I will outline my analysis of the problem. What we are asking for in each case is a pattern in which every face has matching diagonally opposite corner facets and contrasting adjacent corner facets. First, consider any position in which diagonally opposite corner facets match on each face. If we connect every pair of corner cubies that have diagonally opposite corner facets we get two tetrahedrons of corner cubies. A quick examination will show that the four cubies in such a tetrahedron must either be 1) Not all from the same cube, 2) Four copies of the same cubie, or 3) Four different cubies from the same cube, in the correct position and orientation relative to each other. Assuming case 3, we may place one tetrahedron in the home position and consider how the other tetrahedron has been rigidly rotated with respect to the first. There are 12 rotations of the tetrahedron. One is the identity, three are 180-degree rotations about the midpoints of two opposite edges, and six are 120-degree rotations about a vertex. In the identity every face's adjacent corner facets fail to contrast. The 180-degree rotations have the corners like the 3^3 Zig-zag pattern, where two faces have noncontrasting corners. The 120-degree rotations would make checkerboards, but violate the corner twist invariant. Approximate checkerboards can be made from the 180-degree rotation and from the 120-degree rotation with one corner twisted. In the 180-degree approximation, two faces have two wrong facets each. In the 120-degree approximation, three faces have one wrong facet each. Both are achievable with the 2^3 and 4^3, and I think these are as close as you can get on any even-sided cube. SPOTS Spot patterns of the 4^3 are those which have the pattern X X X X X Y Y X X Y Y X X X X X on all nonblank faces. There are a lot of them: every permutation of the centers is possible. There are thus 6! = 720 spot patterns. We may cut this number down by identifying positions that are M-conjugates (recolorings) of each other. As long as I am listing the permutations by conjugacy class, I may as well break them down by recoloring type. Last July I asked a question about the possible rearrangements of colors on the cube. I worked on the solution long enough to find that given a ``standard'' cube, there are five kinds of recoloring up to M-conjugacy. Identity -- The standard coloring. Reflection -- Identity in a mirror. Swap -- Identity with two adjacent colors exchanged. Wrench -- Identity with three adjacent colors cycled. Befuddler -- Wrench in a mirror. The Identity and Reflection are unique colorings. There are twelve Swaps and eight each of the Wrench and Befuddler recolorings. Each recoloring corresponds to 24 face permutations, achievable by whole-cube moves. Here, then, are the spot patterns. Columns correspond to the number of spots in the pattern. The row groupings show the kind of cube coloring the spot pattern comes from. The patterns are given as a permutation of faces: (...XY...) and (Y...X) both mean that face X has a spot colored Y, as shown at the beginning of this message. The number of permutations in each conjugacy class is also given. Number of spots 0 2 3 4 5 6 Coloring -------------------------------------------------------------------- Identity :8 (BUFD):6 (BUL)(FDR):8 (BF)(UD):3 (BF)(UL)(DR):6 -------------------------------------------------------------------- Reflection (BF):3 (BU)(FD):6 (BULFDR):8 (BF)(ULDR):6 (BF)(UD)(LR):1 -------------------------------------------------------------------- Swap (BU):12 (BFU):24 (BFUD):12 (BUFDL):48 (BFULDR):48 (BF)(UL):12 (BF)(UDL):24 (BF)(UDLR):12 (BU)(FDL):48 (BU)(FDLR):48 -------------------------------------------------------------------- Wrench (BUL):16 (BUFL):24 (BFUDL):48 (BFULRD):24 (BU)(FLR):48 (BUL)(FRD):8 (BU)(FLDR):24 -------------------------------------------------------------------- Befuddler (BFUL):48 (BFULD):48 (BFUDLR):16 (BU)(FL):24 (BUFLDR):24 (BFU)(DLR):24 (BU)(FL)(DR):8 -------------------------------------------------------------------- The answer: Fifteen six-spot patterns, six five-spots, eight four-spots, two three-spots, two two-spots, and one no-spot. CROSSES What cross patterns are possible on the 4^3? We must first ask what a cross pattern on the 4^3 should look like. I consider the following two kinds of cross. Thick Cross Thin Cross X O O X X O X X O O O O O O O O O O O O X O X X X O O X X O X X Every thick cross pattern is a rigid rotation or reflection of the edge and face center pieces with respect to the corners. This is just like the 3^3 case except that the 3^3 has face centers that are fixed relative to each other, and so does not allow reflec- tions. So in addition to the thick versions of Plummer's Cross and Cristman's Cross, there are three new crosses. Thick Pons Cross Thick Fliptwist Cross Thick Interlaced Cross U D D U U L L U D D D D B U U B L L L L D D D D U U U U L L L L U D D U U U U U U L L U B U U B L R R L F B B F R L L R L D D L F B B F R U U R R R R R B B B B L L L L U L L U D D D D B B B B U U U U R R R R B B B B L L L L L L L L D D D D B B B B U U U U L R R L F B B F R L L R L L L L L D D L F B B F R U U R U L L U D U U D D R R D U U U U L F F L F D D F R B B R R R R R U U U U F F F F D D D D B B B B R R R R D U U D F F F F D D D D B B B B D R R D L F F L F D D F R B B R B F F B B F F B F F F F D R R D F F F F F F F F R R R R F F F F B F F B R R R R B F F B D R R D For thin crosses, we first examine those in which the arms of the crosses meet at the edges. Again the figure facets are rigidly rotated and reflected with respect to the ground. This time cubie conservation becomes an issue, because of the impossibility of flipping an edge cubie, so there are only three such thin crosses. Thin Pons Cross Thin Plummer Cross Thin Interlaced Cross U U D U U U R U U U D U B B U B U U R U D D D D B B U B R R R R U U D U U U U U U U R U B B U B L L R L F F B F R L R R L L B L F F U F R F R R R R R R B B B B L L L L U U L U B B B B U U U U F F F F L L R L F F B F R L R R U U L U L L B L F F U F R F R R L L R L F F B F R L R R L L L L L L B L F F U F R F R R U U L U D D U D D D L D U U U U L L F L F F D F R B R R L L L L D D U D F F F F D D D D B B B B D D L D D D U D L L F L F F D F R B R R D D L D L L F L F F D F R B R R B B F B B B D B B B F B D D R D B B D B F F F F R R R R D D D D B B F B D D R D B B D B D D R D When we relax the constraint that thin cross arms must meet at the edges, the figure is no longer rigidly transformed with respect to the ground. Indeed, we might expect that adjacent crosses whose arms do not meet might have colors that are opposite on the cube. I carried out a long examination of the cases, and found that this does not happen. In fact, only one new color permutation arises. Thin Fliptwist Cross U U B U B B B B U U B U U U B U L R L L F F U F R L R R B D B B L R L L F F U F L L L L B D B B R R R R U U U U R L R R D D D D L R L L F F U F R L R R B D B B D D F D D D F D F F F F D D F D The only other thin cross patterns in which not all crosses meet at the edges are 43 versions of the Thin Pons Cross (modulo my missing a case or two in the analysis).