12 Pentacube + Tetracube 4-cube: Torsten Sillke, 2000-03-03 Building a 4-cube from the asymmetric pentacubes plus one tetracube. The strategy is to build two 4x4x2 boxes with the 6 pentacubes and a domino such that one box is packed the mirror images of the other. Combining the two halves give solutions for the Tetrominoes 1 1 2 2 1 1 1 1 1 1 1 1 1 1 Combining halves with the domino in differnt position creates solutions for 1 1 1 1 1 1 1 1 The two odd tetracubes could not be used if the pentacubes are in mirror possitions. 1 1 2 2 A 2 2 2 2 2 2 2 2 2 2 2 2 A 1 21 31 33 35 41 71 A 1 21 31 33 36 41 71 A 2 21 31 34 35 41 72 A 2 21 32 33 35 41 72 A 2 21 32 33 36 41 71 A 2 21 32 34 35 41 71 A 1 21 32 34 35 41 72 A 1 21 32 34 36 41 72 A 1 22 31 33 35 41 71 A 2 22 31 33 36 42 72 A 2 22 32 33 35 42 72 A 1 22 32 34 36 41 71 2 2 2 2 B 1 1 2 2 2 2 2 2 2 2 2 2 B 1 21 31 34 35 41 71 B 1 22 31 34 36 41 71 B 4 22 32 33 35 42 72 B 2 22 32 34 35 41 71 B 4 22 32 34 36 42 72 2 2 2 2 D 2 1 1 2 2 2 2 2 2 2 2 2 D 1 21 31 33 35 41 71 D 1 21 32 34 35 42 72 D 1 22 31 33 36 41 71 D 1 22 32 34 36 42 72 2 1 1 2 C 2 2 2 2 2 2 2 2 2 2 2 2 C 1 21 31 33 35 41 71 C 1 21 31 34 35 42 72 C 1 21 32 33 35 41 71 C 1 21 32 34 36 42 71 C 1 21 32 34 36 42 72 C 1 22 31 33 35 41 71 C 1 22 31 33 35 41 72 C 1 22 31 34 36 42 72 C 1 22 32 33 36 41 71 C 1 22 32 34 36 42 72 . 2 2 2 E 2 2 2 2 2 2 2 2 2 2 2 2 E 2 21 31 33 35 41 72 E 4 21 31 33 35 42 71 E 4 21 31 34 35 42 72 E A 21 31 34 36 41 71 E 4 21 31 34 36 42 71 E 2 21 32 33 35 41 71 E 2 21 32 33 35 41 72 E 4 21 32 33 35 42 71 E 2 21 32 33 36 41 71 E 2 21 32 33 36 41 72 E 2 22 31 34 35 42 71 E 2 22 31 34 35 42 72 E 4 22 31 34 36 41 72 E 2 22 31 34 36 42 71 E 2 22 31 34 36 42 72 E 4 22 32 33 35 41 72 E A 22 32 33 35 42 72 E 4 22 32 33 36 41 71 E 4 22 32 34 36 41 72 E 2 22 32 34 36 42 71 2 . 2 2 F 2 2 2 2 2 2 2 2 2 2 2 2 F 2 21 32 33 35 41 71 F 1 21 32 33 35 41 72 F 1 22 31 34 36 42 71 F 2 22 31 34 36 42 72 2 2 2 2 G impossible 2 . 2 2 2 2 2 2 2 2 2 2 * * * - -A * * * - * * * * * * * * * * * * * * * * * * * * * * * * -A 1 21 31 33 35 42 72 -A 2 21 31 34 36 41 71 -A 2 21 32 34 35 41 71 -A 1 21 32 34 36 42 72 -A 1 22 31 33 35 42 71 -A 1 22 31 33 36 42 71 -A 2 22 31 33 36 42 72 -A 2 22 31 34 35 42 72 -A 2 22 31 34 36 42 71 -A 2 22 32 33 36 42 71 -A 1 22 32 34 35 42 72 -A 1 22 32 34 36 42 72 * * - * -B * * - * * * * * * * * * * * * * * * * * * * * * * * * * -B 4 21 31 33 35 41 71 -B 2 21 31 33 36 42 72 -B 4 21 31 34 36 41 71 -B 1 21 32 33 35 42 72 -B 1 22 32 33 36 42 72 Which combination build which figure how often? 21 31 33 35 41 71 A1 -B4 C1 D1 21 31 33 35 41 72 E2 21 31 33 35 42 71 E4 21 31 33 35 42 72 -A1 21 31 33 36 41 71 A1 21 31 33 36 42 72 -B2 21 31 34 35 41 71 B1 21 31 34 35 41 72 A2 21 31 34 35 42 72 C1 E4 21 31 34 36 41 71 -A2 -B4 EA 21 31 34 36 42 71 E4 21 32 33 35 41 71 C1 E2 F2 21 32 33 35 41 72 A2 E2 F1 21 32 33 35 42 71 E4 21 32 33 35 42 72 -B1 21 32 33 36 41 71 A2 E2 21 32 33 36 41 72 E2 21 32 34 35 41 71 +-A2 21 32 34 35 41 72 A1 21 32 34 35 42 72 D1 21 32 34 36 41 72 A1 21 32 34 36 42 71 C1 21 32 34 36 42 72 -A1 C1 22 31 33 35 41 71 A1 C1 22 31 33 35 41 72 C1 22 31 33 35 42 71 -A1 22 31 33 36 41 71 D1 22 31 33 36 42 71 -A1 22 31 33 36 42 72 +-A2 22 31 34 35 42 71 E2 22 31 34 35 42 72 -A2 E2 22 31 34 36 41 71 B1 22 31 34 36 41 72 E4 22 31 34 36 42 71 -A2 E2 F1 22 31 34 36 42 72 C1 E2 F2 22 32 33 35 41 72 E4 22 32 33 35 42 72 A2 B4 E4 22 32 33 36 41 71 C1 E4 22 32 33 36 42 71 -A2 22 32 33 36 42 72 -B1 22 32 34 35 41 71 B2 22 32 34 35 42 72 -A1 22 32 34 36 41 71 A1 22 32 34 36 41 72 E4 22 32 34 36 42 71 E2 22 32 34 36 42 72 -A1 B4 C1 D1 Therefore the most valuable combinations are: 21 31 33 35 41 71 A1 -B4 C1 D1 21 31 34 36 41 71 -A2 -B4 EA 21 32 33 35 41 71 C1 E2 F2 21 32 33 35 41 72 A2 E2 F1 The odd Tetracubes: ------------------- Packing a symmetric figure with one tetracube plus two pentacubes. 1 2 1 1 2 2 1 1 2 2 2 2 is packable with 2 1 plus (71, 72) Unhappyly this cannot be extended with two symmetric halve cubes. All possible combinations use either 71 or 72. . . 2 2 . 1 2 2 2 2 2 2 2 2 2 2 21 31 34 42 72 22 31 35 42 71 22 32 35 42 72 21 31 36 41 71 21 32 36 41 72 22 32 33 41 71 . . 2 2 1 . 2 2 2 2 2 2 2 2 2 2 31 34 36 42 71 31 34 35 41 71 21 31 34 41 71 21 32 34 41 72 2 . . 2 2 1 . 2 2 2 2 2 2 2 2 2 22 31 33 42 72 - - - - - - - - - - - - - - - - - - - - - - - - - - - - Plans: 21 31 33 35 41 71 : A1 -B4 C1 D1 A 71 31 31 33 71 31 33 33 71 71 35 35 41 41 41 21 . . 31 31 71 35 35 33 41 21 35 33 41 21 21 21 B 21 71 71 71 21 71 35 33 21 21 35 33 41 41 31 31 21 . 71 35 41 . 35 35 41 31 33 33 41 31 31 33 21 71 71 71 21 71 35 33 21 21 35 35 41 41 31 31 21 . 71 33 41 . 33 33 41 31 33 35 41 31 31 35 21 31 31 35 21 31 33 35 21 21 33 33 41 41 71 33 21 . 31 31 41 . 35 35 41 71 35 33 41 71 71 71 21 31 31 33 21 31 33 33 21 21 35 35 41 41 71 35 21 . 31 31 41 . 35 33 41 71 35 33 41 71 71 71 C 71 41 41 41 71 71 31 31 71 35 33 31 21 35 33 33 71 . . 41 21 21 35 41 21 35 35 31 21 33 33 31 D 41 31 31 33 41 31 33 33 41 35 35 21 71 21 21 21 41 41 31 31 71 . . 33 71 71 35 33 71 21 35 35