Results of analysis of Gosper's 7x7x7 modified puzzle version 1 (probably final), June 2, 2015 Pieces in Bill's original and modified versions are: size original modified 1x3x3 3 3 1x2x2 3 0 1x2x3 0 2 1x2x4 10 10 1x4x4 14 14 Solutions counted here are distinct with respect to the 48 rotations and reflections of the assembled cube. The three 1x3x3 pieces are named 1 through 3. The two 1x2x3 pieces are named 4 and 5. The 10 1x2x4 pieces and the 14 1x4x4 pieces are named a through x. Within these three categories, which piece has which name can vary. (A given lower case letter in a given diagram may name either a 1x2x4 or a 1x4x4.) There are three types of solution, and each type has subtypes. In type 1 solutions, two of the 1x3x3 pieces lie in parallel planes. The three 1x3x3 pieces lie in orthogonal planes in types 2 and 3. In type 2 solutions, one of the planes oddified (receiving an odd number of cells) by the 1x2x3 pieces is on the surface of the 7x7x7. In type 3 solutions, the planes oddified by the 1x2x3 pieces are not on the surface of the 7x7x7. For each type, the subtypes are distinguished by the positions of the 1x2x3 pieces. TYPE 1 SOLUTIONS In type 1, it turns out that all solutions have the three 1x3x3 pieces and one 1x2x3 piece in the same place, shown below. . . . . . . . layer 0 -- type 1 . . . . . . . . . . . . . . . . . . . . . 4 4 4 . . . . 4 4 4 . . . . 1 1 1 . . . . . . . . . . . layer 1 W W W 3 X X X . . . 3 . . . . . . 3 . . . . . . . . . . Y Y Y . Z Z Z 1 1 1 . . . . . . . . . . . layer 2 W W W 3 X X X . . . 3 . . . . . . 3 . . . . . . . . . . Y Y Y . Z Z Z 1 1 1 . . . . . . . . . . . layer 3 . . . 3 . . . . . . 3 . . . . . . 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 2 layer 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 2 layer 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 2 layer 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The 1x3x3 pieces 1 and 2 are in parallel planes. Piece 4 is always in the place shown in layer 0. The other 1x2x3, piece 5, can be in the position shown by letters W, X, Y or Z (upper case; lower case are 1x2x4 or 1x4x4 pieces). The other three of those positions, and all the dot cells, are occupied by 1x2x4 and 1x4x4 pieces. When coordinate axes are mentioned, x goes from 0 (left) to 6 (right); y goes from 0 (first row) to 6 (last row) within a layer; and z goes from 0 (layer 0, base of the cube) to 6 (layer 6, top of the cube). The subtypes of type 1 are defined by where the second 1x2x3, piece 5, is placed. For each of the 4 positions, the number of solutions is: W 8066 = 2 * 37 * 109 X 3912 = 2 * 2 * 2 * 3 * 163 Y 34121 = 149 * 229 Z 12146 = 2 * 6073 ----- 58245 distinct type 1 solutions TYPE 2 SOLUTIONS In all type 2 solutions, the 1x3x3 pieces are as shown below. Letters A, B, C and D designate positions that one 1x2x3 may occupy, and W, X, Y and Z are positions that the other 1x2x3 may occupy. Each 1x2x3 piece oddifies an outer layer and the layer next to it. A A A . B B B layer 0 -- type 2 W W W . X X X . . . . . . . . . . . . . . C C C . D D D Y Y Y . Z Z Z . . . . 2 2 2 A A A . B B B layer 1 W W W . X X X . . . . . . . . . . . . . . C C C . D D D Y Y Y . Z Z Z . . . . 2 2 2 . . . . . . . layer 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 2 . . . . . . . layer 3 . . . . . . . . . . . . . . . . . 3 . . . . . . 3 . . . . . . 3 . . . . . . . . . . . . . . . . . layer 4 . . . . . . . . . . . . . . . . . 3 . . . . . . 3 . . . . . . 3 . . . . . . . . . . . . . . . . . layer 5 . . . . . . . . . . . . . . . . . 3 . . . . . . 3 . . . . . . 3 . . . . . . . . . . 1 1 1 . . . . layer 6 1 1 1 . . . . 1 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Only 12 of the possible combinations of positions for the 1x2x3 pieces occur in solutions. They, and the number of solutions they have, are: A & W 45791 = 29 * 1579 A & X 21534 = 2 * 3 * 37 * 97 A & Y 14743 = 23 * 641 A & Z 16077 = 3 * 23 * 233 B & W 14860 = 2 * 2 * 5 * 743 B & X 45791 = 29 * 1579 B & Y 7262 = 2 * 3631 B & Z 23564 = 2 * 2 * 43 * 137 C & Y 26338 = 2 * 13 * 1013 C & Z 11508 = 2 * 2 * 3 * 7 * 137 D & Y 6619 = 6619 D & Z 34121 = 149 * 229 ------ 268208 distinct type 2 solutions In 4 of the above subtypes, such as A&W, the two 1x2x3 pieces are face to face. The two-piece assembly can be rotated so that one piece lies entirely in layer 0, and the other entirely in layer 1. It might appear that the analysis missed, or mistakenly suppressed, that configuration. However, it is equivalent to another configuration in this report. It is counted as distinct. In this analysis, re-packings, re-orientations or mirroring of parts of a solution, such as this 2x2x3 unit, are NOT considered to be equivalent. Doing so would open the door to a large challenge to define how complex a unit can be, and how complicated a re-packing can be, before solutions should be counted as distinct. The rotation of the 2x2x3 unit is discussed more below. TYPE 3 SOLUTIONS In all type 3 solutions, the 1x3x3 pieces are in the same positions as in type 2. But the 1x2x3 pieces oddify two layers between an outside layer and the middle layer parallel to that outer layer. . . . . . . . layer 0 -- type 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 2 . . . . . . . layer 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 2 . . . . . . . layer 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 2 . . . . . . . layer 3 . . . . . . . . . . . . . . . . . 3 . . . . . . 3 . . . . . . 3 . . . . . . . . . . A A A . B B B layer 4 W W W . X X X . . . . . . . . . . 3 . . . C C C 3 D D D Y Y Y 3 Z Z Z . . . . . . . A A A . B B B layer 5 W W W . X X X . . . . . . . . . . 3 . . . C C C 3 D D D Y Y Y 3 Z Z Z . . . . . . . 1 1 1 . . . . layer 6 1 1 1 . . . . 1 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Only 10 of the possible combinations of positions for the 1x2x3 pieces occur in solutions. They, and the number of solutions they have, are: A & W 34121 = 149 * 229 A & X 12146 = 2 * 6073 A & Y 8066 = 2 * 37 * 109 A & Z 3912 = 2 * 2 * 2 * 3 * 163 B & W 8688 = 2 * 2 * 2 * 2 * 3 * 181 B & X 26338 = 2 * 13 * 1013 B & Y 892 = 2 * 2 * 223 B & Z 9306 = 2 * 3 * 3 * 11 * 47 C & Y 15502 = 2 * 23 * 337 D & Z 15502 = 2 * 23 * 337 ------ 134473 distinct type 3 solutions GRAND TOTAL 58245 distinct type 1 solutions 268208 distinct type 2 solutions 134473 distinct type 3 solutions ------ 460926 grand total of distinct solutions 1-TO-1 MAPPINGS The reader has probably noticed that some solution counts reappear in the results above. Seven counts appear twice, and one (34121) appears thrice. In the following discussion, the 34121 case is divided into two, flagged with an asterisk. 8066 type 1 W, type 3 A&Y 3912 type 1 X, type 3 A&Z 34121* type 1 Y, type 3 A&W 12146 type 1 Z, type 3 A&X In type 1, piece 1 and piece 4 form a V-shaped unit that can be flipped, changing the the configuration to type 3. This explains the four pairs above. This flipping was first announced by Bill in email on 3/31/2013. That email is discussed further, under "philosophy and program design". 26338 type 2 C&Y, type 3 B&X 34121* type 2 D&Z, type 3 A&W These two pairings have a clear and interesting equivalence, related to rotating the 2x2x3 unit (made of two 1x2x3 pieces side by side) on its long axis. Take a type 2 C&Y solution. Rotate the 2x3x3 by 1/4 turn. Then rotate the entire cube 1/4 turn around y=z=3, so that the bottom (layer 0) becomes the side close to you (y=6). Then rotate the cube 1/2 turn around x=y=3, so piece 2 is now where piece 1 was originally. Except for a change of piece names, this is a type 3 B&X solution. The correspondence between type 2 D&Z and type 3 A&W is similar. 45791 type 2 A&W, type 2 B&X This correspondence also hinges on rotating a 2x2x3. Take a type 2 A&W solution. Rotate the A&W unit. Then perform the whole-cube 1/4 and 1/2 turn rotations described above. The result is a type 2 B&X solution. 15502 type 3 C&Y, type 3 D&Z Again, take a type 3 C&Y solution. Rotate the C&Y unit. Perform the whole-cube 1/4 and 1/2 turn rotations as above. This makes a type 3 D&Z solution. ONE MORE 1x4x4 PIECE A search was run with 8 (instead of 10) 1x2x4 pieces and 15 (instead of 14) 1x4x4 pieces. Just as with Bill's original 7x7x7 puzzle, there are no solutions with this change. PHILOSOPHY AND PROGRAM DESIGN Puzzle analysis can result in useful insights, such as, "these pieces must be in this position because ...". Or, the result can be merely a report of what a search found, such as, "all the solutions happen to have these pieces in position so-and-so". The former is generally more interesting, and the latter is usually more concise. The paragraphs above tend to the latter, dry factoids. But the search program was designed using various insights, without which the run time would have been impractically huge. The first is of course the parity constraint that makes human solution of the puzzle feasible. The second design principle was to list the ways each plane in a group of 7 parallel planes can be oddified by the 1x3x3 and 1x2x3 pieces. Then, to list the ways that all three groups of 7 such planes can be so oddified simultanelously. This cuts down drastically on the space to be searched, but still generates a large number of cases that the program finds have no solution. The third design principle is a 4-coloring of the 7x7x7 cube. The coloring can be defined by: any 1x2x2, in any position or orientation within the cube, contains one cell of each color. There is an excess of the color of the cube's 8 corner cells. Only a fraction of the possible placements of the 1x3x3 and 1x2x3 pieces satisfy the required color tallies. This cuts down significantly on the search space, although total search run time is still a few days. (The program design has three "classes" of solution. Class 3 initially seemed like it might have many solutions, but 4-coloring eliminated them all, obviating the need to place 1x2x4 and 1x4x4 pieces in any of the class 3 5-piece skeletons! In class 3, as in types 2 and 3, the 1x3x3 pieces lie in mutually orthogonal planes. But in class 3, the planes oddified by one 1x2x3 are perpendicular, not parallel, to those oddified by the other 1x2x3. Note that this is NOT equivalent to the 1x2x3 pieces lying in perpendicular planes. For example, they can lie in the very same plane, but one rotated 1/4 turn.) An easy observation is that in all solutions, the 3-long axis of each 1x2x3 piece is perpendicular to the "middle" (all interior) 1x3x3. It is not clear whether some not-too-complicated reason explains this, or whether it just happens to be so. A similar conundrum arises with Bill's original 7x7x7 puzzle. Bill's 3/31/2013 email noting the flipping of a V-shaped unit was mentioned above. A further statement in that email was that you can maneuver face-to-face 1x2x3 pieces [in a type 2 or type 3 solution] to adjoin a 1x3x3, thus making that V-shaped unit. That is, there are manipulations that link all A&W, B&X, C&Y and D&Z sub-types in both type 2 and type 3. This linking could be said to generate one large family of solutions, all transformable into each other. The results reported here are compatible with there being such manipulations, but do not say whether they exist. I would guess that showing that such manipulations can always be done, would hinge on where some 1x2x4 and 1x4x4 pieces must go. And those placements could be justified either by "an exhaustive search shows this piece is always here, in every solution", or by "this piece must be here because..." -- returning to the philosophy question of how to state analysis results. The assertion of a large family of solutions (type 1, and all type 2 and 3 with face-to-face 1x2x3 pieces) suggests a further question. With about the same complexity of manipulations that link that family's members, can the additional solutions in type 2 and type 3 be brought into the family? That is, can similar manipulations transform the not-face-to-face solutions into face-to-face solutions? Bill's original 7x7x7's solutions are all one such family, I found. For the modified version, this is currently an open question.