Date: 27 January 1981 0102-EST (Tuesday) From: Jim Saxe, Dan Hoey To: Cube-Lovers at mit-mc Subject: Pretty Patterns and Solutions Sender: Dan Hoey at CMU-10A Reply-To: Dan Hoey at CMU-10A Message-Id: <27Jan81 010221 DH51@CMU-10A> We are disappointed at Chris C. Worrell's use of the term "Baseball" for the position known in the literature as the "Worm". Worrell's term propagates the apparently popular misconception that baseballs are covered with three-lobed pieces of leather. The position which *we* call "Baseball" reflects the construction much more accurately: D D D U U U F F F U U U L R B R B L B L R F F F L R B R B L B L R D D D L R B R B L B L R F F F D D D U U U Currently, our best process for this position is 34 qtw. The corners are fixed with FRLUDB, two edge four-cycles are inserted in the middle, and a Spratt wrench is conjugated inside that: FRL (RRLL UUDD F' U (F' LUD' BUD' RUD' FUD' F) UDD RRLL F) UDB. The class of patterns which Bill Vaughan calls "Swirl Patterns" (15 January 1981 19:13 cst) are also known as "6-2L" patterns in Singmaster, and the particular one he calls the "Pinwheel" (on 16 January 1981 12:09 cst) is an M-conjugate of the AC-symmetric "Twelve-L's" mentioned in our message on Symmetry and Local Maxima (14 December 1980 1916-EST, Section 6). [Incidentally, the diagram he displays in the message of 16 January is in error; the left and right face centers have been swapped. This is made less obvious by the unusual orientation of the cube in that diagram.] We have found a totally magical 12 qtw process for the Pinwheel: FB LR F'B' U'D' LR UD. Vaughan's definition of Swirl Patterns seems unduly restrictive to us on one count: he requires the two L's on each face to be of "complementary" (evidently meaning opposite) colors. This is not necessary for an L pattern. According to our analysis, however, at least two of the faces of any L pattern must have L's of opposite colors, and five is easily seen to be impossible. We know of no patterns having three or four such pairs. But there are several with two pairs. Our favorite example is a relative of the Baseball which we name for Linda Lue Leiserson, who has the appropriate initials. D D D F U D F F F U U U L B B R L L B R R D F U L R B R B L B L R D D D L L B R R L B B R F F F U D F U U U We have a 24 qtw process for Linda Lue's L: F'BB L (B U'LR' F'LR' D'LR' B'LR' B') R UD B'FF This has a Spratt wrench, conjugated by B', embedded in magic. David C. Plummer (3 SEP 1980 2123-EDT) reported that it is possible for each of the six faces of the cube to show a capital "T". Our analysis indicates that there are two sorts of T patterns: D U D D D U D U D U U U U U U D D U L L L F F F R R R L R R F F F R L L R L R B F B L R L L L L B F B R R R R L R B F B L R L L R R B F B R L L D D D D U U U D U D D D U D U D U U B B B B B B F B F F B F F B F F B F Tanya's T Plummer's T Tanya's T is named for Tanya Sienko (who inspired the problem) and for euphony. Plummer's T is named for Plummer's Cross (which has the same symmetry group) and for homophony. There are 24 M-conjugates of Tanya's T, while Plummer's T has 8 M-conjugates. By adapting a process due to David C. Plummer, we have developed a 16-qtw process for Tanya's T: (FF UU)^3 (UU LR')^2. The first part swaps two pairs of edge cubies, and the second part is magic. We have found a 28-qtw process for Plummer's T, which is entirely magical: FF UD' F'B' RR F'B U'D RL FF RL' UD' RL FF R'L U'D'. A position which is not so visually striking, but which is important in the symmetry theory we have discussed earlier, is "All Corners Twisted": B U B U U U F U F U L U L F R U R U R B L L L L F F F R R R B B B D L D L F R D R D R B L F D F D D D B D B This can be achieved in 30 qtw with FLU (LRRFFB')^4 U'L'F'. The process is a conjugated from a 24 qtw process invented by Thistlethwaite. Unfortunately, Thistlethwaite's process twists the wrong corners, and no cancellation can be performed in the conjugation. If any process can be found which twists four corners clockwise and four counterclockwise, leaving the rest of the cube fixed, then any such pattern can be made by adding at most 6 qtw.