prime boxes of (71) complete list Torsten Sillke, BI, 15.04.93 1 (boxes completed 07.02.93) 1 1 2 2*5: 2p, 4, 6, ... 2n 3*5: 6p, 8p, 9p, 10p, 11p, 12, 13p, ... 8..13 + 6n 4*5: 2, 4, 5p, ... 4..5 + 2n 5*5: 4p, 5p, 6, 7p, ... 4..7 + 4n 6*5: 2, 3p, ... 2..3 + 2n 7*5: 4, 5p, 6, 7p, ... 4..7 + 4n 8*5: 2, 3p, ... 2..3 + 2n k*5: 2 (2*5*k), 3 (3*5*k), n>3 (k=4..8 + 4m) // k > 8 2*10: 2, 4, 5, ... 4..5 + 2n 3*10: 4p, 5p, 6, 7p 2*15: 2, 4, 6, 7p 3*15: 4p, 5, 6, 7p 2*2*5: 1p, ... 2*3*5: 2, 3p, ... 2..3 + 2n 3*3*5: 2p, 3p, ... 2..3 + 2n Impossible: 2*3*5n 3*3*5n 2*5*u (u odd) 3*5*4, 3*5*5, 3*5*7 Annotations: the 2x2x5 box: (unique) 3 2 2 times 2 3 gives 2x2x5. 2x5xN dissects into 2x2x5 boxes. Therefore 2x5xu (with u odd) is impossible. the 3x5x6 box: (unique) 25 26 20 15 12 13 26 26 26 15 13 13 27 22 22 15 15 13 27 27 22 16 16 16 27 24 22 17 16 11 25 25 20 18 12 12 26 21 20 15 14 10 27 21 21 14 14 13 23 21 22 17 14 16 24 24 19 17 11 11 25 20 20 18 12 10 25 21 18 18 12 10 23 23 19 18 10 10 23 24 19 19 14 11 23 24 19 17 17 11 the 3x5x8 box: (unique) 31 27 27 27 22 15 15 15 31 31 27 22 22 15 12 12 31 32 23 24 22 19 13 12 32 32 24 24 19 19 19 12 33 32 25 24 20 20 14 14 31 28 23 27 16 17 15 10 29 29 23 23 17 17 13 13 30 29 23 21 22 17 13 12 33 29 25 26 19 20 13 14 33 32 25 24 18 20 11 14 28 28 28 21 16 16 16 10 30 30 28 21 21 16 10 10 30 29 26 21 18 17 11 10 30 26 26 26 18 18 11 11 33 33 25 25 18 20 11 14 the 3x5x9 box: (unique) 34 34 34 24 21 18 18 13 10 35 34 28 24 24 22 18 13 13 36 30 28 24 22 22 18 13 11 36 36 28 28 25 22 15 15 15 36 33 29 25 25 25 15 14 14 35 31 34 24 21 21 21 13 10 35 30 30 26 19 21 18 16 10 36 30 28 23 23 23 17 11 11 32 30 29 27 23 22 20 15 14 33 33 29 25 20 20 20 12 14 31 31 31 26 19 19 16 10 10 35 35 31 26 19 16 16 16 11 32 32 26 26 19 23 17 12 11 32 33 27 27 27 17 17 12 12 32 33 29 29 27 20 17 12 14 the 3x5x10 box: 2 times part A 37 38 32 27 27 2 2 2 2 2 <- part A (unique) 38 38 38 28 28 2 2 2 2 2 39 34 34 34 28 2 2 2 2 2 39 39 34 31 28 2 2 2 2 2 39 36 31 31 31 2 2 2 2 2 37 37 32 27 25 2 2 2 2 2 38 33 32 27 2 2 2 2 2 2 39 33 33 34 28 2 2 2 2 2 35 33 30 26 26 26 2 2 2 2 36 36 31 26 29 2 2 2 2 2 37 32 32 27 25 2 2 2 2 2 37 33 30 25 25 2 2 2 2 2 35 35 30 30 25 2 2 2 2 2 35 36 30 29 26 2 2 2 2 2 35 36 29 29 29 2 2 2 2 2 the 3x5x11 box: 40 40 32 32 32 25 23 19 19 19 13 41 40 33 32 34 25 25 19 17 13 13 42 40 34 34 34 25 22 20 20 20 13 42 42 36 35 29 26 22 20 15 15 15 42 39 35 35 35 26 22 22 15 14 14 41 37 33 33 32 25 23 21 19 11 10 41 40 33 30 27 27 23 17 17 11 11 42 36 33 34 28 27 24 18 20 11 13 38 36 36 36 29 27 22 18 16 15 14 39 39 35 31 29 26 26 18 18 12 14 37 37 37 30 28 23 23 21 17 11 10 41 41 37 30 28 28 21 21 17 10 10 38 38 30 30 28 27 24 21 16 12 10 38 39 31 29 29 24 24 18 16 12 12 38 39 31 31 31 26 24 16 16 12 14 the 3x5x13 box: (part A and part B) 31 29 24 24 19 17 12 12 <- part B (unique) 31 29 25 24 19 17 13 13 31 31 26 24 19 19 18 13 32 32 33 22 22 22 16 13 33 33 33 22 21 16 16 16 29 29 25 23 17 17 12 10 31 27 25 24 19 18 12 30 27 26 26 18 18 18 13 32 27 27 20 22 15 11 11 11 32 33 28 21 21 16 11 14 30 29 23 23 23 17 12 10 30 30 25 25 23 15 10 10 30 27 26 20 20 15 15 10 32 28 26 20 21 15 14 11 28 28 28 20 21 14 14 14 the 3x4x10 box: (unique) 31 31 26 26 26 21 16 16 11 10 32 32 27 26 21 21 21 16 11 10 33 32 27 24 24 24 20 16 11 11 33 32 27 27 24 20 20 20 12 12 31 28 28 28 26 18 18 18 10 10 31 28 29 23 21 19 18 16 11 13 30 32 27 25 22 24 17 14 15 12 33 33 25 25 25 20 15 15 15 12 31 29 28 23 23 19 19 18 13 10 29 29 29 23 22 19 17 13 13 13 30 30 30 23 22 19 17 14 14 14 33 30 25 22 22 17 17 15 14 12 the 3x4x15 box: 2 times part C 31 31 32 26 20 20 20 19 2 2 2 2 2 2 2 <- part C (unique) 32 32 32 24 20 19 19 19 2 2 2 2 2 2 2 33 33 33 24 24 21 17 2 2 2 2 2 2 2 2 33 30 27 24 21 21 21 2 2 2 2 2 2 2 2 31 28 26 26 26 20 16 2 2 2 2 2 2 2 2 31 32 27 24 26 22 19 2 2 2 2 2 2 2 2 29 33 27 27 23 17 17 18 2 2 2 2 2 2 2 30 30 27 25 21 18 18 18 2 2 2 2 2 2 2 31 28 28 28 22 16 16 16 2 2 2 2 2 2 2 29 29 28 22 22 22 17 16 2 2 2 2 2 2 2 29 30 25 23 23 23 17 2 2 2 2 2 2 2 2 29 30 25 25 25 23 18 2 2 2 2 2 2 2 2 the 5x5x4 box: 28 28 28 14 15 28 21 21 15 15 29 22 22 22 15 29 23 22 17 11 29 29 17 17 17 26 28 21 14 14 26 26 21 12 12 26 23 23 22 15 29 23 19 13 11 27 23 17 13 11 26 20 16 14 12 25 20 21 14 12 27 20 20 18 10 27 27 19 11 11 27 24 19 13 13 25 25 16 16 16 25 20 18 16 12 25 18 18 18 10 24 19 19 10 10 24 24 24 13 10 the 5x5x5 box: 4 solutions x x 20 20 20 x x 21 20 12 34 31 22 14 11 34 31 22 14 14 34 34 32 14 13 x x 21 21 20 x x 21 12 12 31 31 21 14 11 34 32 22 22 11 32 32 32 13 13 x x 19 15 12 x x 19 19 12 29 31 19 11 11 29 29 22 24 13 29 24 24 24 13 x x 19 15 10 x x 23 15 17 29 26 17 17 17 26 26 18 18 18 27 26 24 18 16 x x 15 15 10 x x 23 10 10 27 23 23 17 10 27 27 23 16 18 27 26 16 16 16 the 5x5x7 box: 5 5 5 5 5 is unique (-> 5*5*7) 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 36 32 25 25 25 . . 36 36 26 25 16 . . 36 35 27 20 16 16 12 . . 27 28 16 12 12 . . 28 28 28 19 12 36 32 26 23 25 . . 35 32 26 20 16 . . 35 35 35 20 15 15 15 . . 27 27 19 15 14 . . 28 19 19 19 12 32 32 23 23 23 . . 33 34 26 26 23 . . 34 34 34 20 20 14 15 . . 27 24 14 14 14 . . 24 24 24 11 11 31 29 29 18 17 . . 33 33 29 18 18 . . 34 30 29 18 13 13 13 . . 22 22 22 13 11 . . 24 22 21 10 11 31 31 31 18 17 . . 33 31 29 17 17 . . 33 30 30 30 17 10 13 . . 30 21 22 10 10 . . 21 21 21 10 11 the 5x7x7 box: 5 5 5 5 5 5 5 (-> 5*7*7) 5 5 5 5 5 5 5 and 5 5 5 5 5 5 5 the first has 2 solutions 5 5 5 5 5 5 5 5 5 5 5 5 5 5 the second has 5 solutions 5 5 5 5 5 5 5 5 5 5 5 5 5 5 prime boxes of (71/72) (with reflections) 2*5: 2p, 4, 6, 7p, ... 3*5: 4p, 5p, 6p, 7p, ... 4*5: 2, 3p, ... 5*5: 3p, 4p, 5p, ... 2*10: 2, 3p, ... 3*10: 2p, 3p, ... 2*15: 2, 3p, ... 3*15: 2p, 3p, ... 2*2*5: 1p, ... 2*3*5: 2, 3p, ... 2..3 + 2n 3*3*5: 2p, 3p, ... 2..3 + 2n Impossible: 2*3*5, 2*5*5, 3*3*5 Annotations: the hyperboxes 2*3*3*5 and 3*3*3*5 can be made with: 2 2 2 4 4 4 2 2 2 2 2 3 4 4 3 3 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 3 3 3 3 3 3 3 3 3 3 (6) (3) (5) <- number of solutions The excess from the 3*3*5 box can be transformed in the 4th dimension in each case. (There are suitable solutions.) As two 71 build a x-pentomino of height 2, one can try to build a 3*3*5 box with 71 and x to get a 2*3*3*5 hyperbox. But 2*3*5, 2*5*5 and 3*3*5 are still impossible, if you add the x-pentomino.