The 9th Pentacube Contest 97/98 and CFF Contest 33. Torsten Sillke, 25.04.98, Frankfurt/M Build the figure "top of tower" with all 29 pentacubes. Each number says, how 1 2 3 4 3 2 1 many cubes are stacked. 2 3 4 5 4 3 2 3 4 5 6 5 4 3 This top of a tower consits of 175 cubes. 4 5 6 7 6 5 4 To build it up with the 145 cubes of the 29 3 4 5 6 5 4 3 pentacubes leaves you a space of 30 cubes 2 3 4 5 4 3 2 to be placed in the interior of the structure. 1 2 3 4 3 2 1 You may locate these empty cubes at any place you want. But there is one condition: The structure must be selfsupporting and resist even a strong pressure on the top of it. Solution of partition number 7 (empty cubes are marked with 'X'): 42 42 X . 61 22 X . 13 71 X . 60 11 X . X dissect the 42 36 72 X 61 22 21 X 13 90 90 X 60 11 81 X X 50 X problem into 42 72 72 72 22 22 21 70 13 80 80 80 82 82 82 12 10 the five parts: 36 42 X . 61 22 X . 13 71 X . 60 11 X . X A A A D D D D 36 36 41 X 61 61 21 X 71 71 90 X 60 82 81 X 50 50 50 A A A D D D D . 41 41 72 . 70 70 70 . 71 80 40 . 82 12 12 10 A A A E D D D A A E E E C C 36 32 X . 30 30 X . 13 34 X . 60 11 X . X B B B E C C C . 32 41 X . 21 21 X . 90 90 X . 81 81 X 35 50 X B B B B C C C . . 31 31 . . 70 33 . . 80 40 . . 81 12 10 B B B B C C C . 32 32 . . 30 30 . . 34 34 . . 11 37 . 35 . . 41 31 . . 33 33 . . 34 40 . . 37 37 35 35 20 . . . 31 . . . 33 . . . 40 . . . 12 10 . . 32 . . . 30 . . . 34 . . . 37 . 35 51 . . . 31 . . . 33 . . . 40 . . . 37 20 20 20 -> 20 51 51 -> 51 . . . . . . . . . . . . . . . . 10 51 Partitions for A, B, C, D and E=(10,20,35,50,51) 1) 32 34 41 42 71 90: 1 31 36 40 60 70 72: 1 21 30 33 61 80 81: 1 11 12 13 22 37 82: 1 2) 31 40 60 61 72 80: 1 30 33 36 41 42 70: 1 21 32 34 71 81 90: 1 11 12 13 22 37 82: 1 3) 31 32 41 42 60 81: 1S 30 34 36 61 71 90: 1 21 33 40 70 72 80: 1 11 12 13 22 37 82: 1 4) 30 33 41 42 70 82: 1 22 32 34 60 61 90: 1 13 21 31 40 71 80: 1 11 12 36 37 72 81: 1 5) 22 31 32 42 60 61: 1 21 30 33 41 70 82: 1 13 34 40 71 80 90: 1 11 12 36 37 72 81: 1 6) 31 32 36 41 42 72: 1 21 30 33 61 80 81: 1 13 22 34 40 70 90: 1 11 12 37 60 71 82: 1 7) 31 32 36 41 42 72: 1 21 22 30 33 61 70: 1 13 34 40 71 80 90: 1 11 12 37 60 81 82: 1S 8) 30 33 36 41 42 70: 1 22 32 34 60 61 90: 1 13 21 31 40 71 80: 1 11 12 37 72 81 82: 1 9) 31 34 36 42 71 90: 1 21 33 40 60 80 81: 1 12 22 37 41 72 82: 1 11 13 30 32 61 70: 1 10) 22 32 34 60 61 90: 1 21 30 33 41 70 82: 1 12 36 37 42 71 81: 1 11 13 31 40 72 80: 1 11) 32 34 41 42 60 82: 1 21 22 30 33 61 70: 1 12 36 37 71 81 90: 1 11 13 31 40 72 80: 1 12) 22 32 34 41 60 61: 1 21 30 33 42 70 82: 1 12 36 37 71 81 90: 1 11 13 31 40 72 80: 1 13) 22 32 34 60 61 90: 1 21 30 33 36 42 70: 1 12 37 41 71 81 82: 1 11 13 31 40 72 80: 1 14) 31 34 41 60 72 90: 1 21 30 33 61 80 81: 1 12 22 36 37 71 82: 1 11 13 32 40 42 70: 1 15) 31 34 60 61 81 90: 1 21 30 33 36 42 70: 1 12 22 37 41 72 82: 1 11 13 32 40 71 80: 1 16) 32 33 36 42 60 90: 1 21 34 40 71 80 82: 1 12 22 30 31 70 81: 1 11 13 37 41 61 72: 1 17) 30 33 42 70 72 82: 1 13 34 40 71 80 90: 1 12 36 37 41 61 81: 1 11 21 22 31 32 60: 1S 18) 32 34 60 70 72 90: 1 13 21 31 40 71 80: 1 12 36 37 41 81 82: 1 11 22 30 33 42 61: 1 19) 30 34 36 41 72 90: 1 13 21 31 40 71 80: 1 12 37 42 60 61 82: 2 11 22 32 33 70 81: 1 20) 22 33 40 42 70 90: 1 13 21 32 34 60 72: 1 12 36 37 71 81 82: 1 11 30 31 41 61 80: 1 21) 32 34 36 42 60 82: 1 21 33 40 70 72 80: 1 12 13 22 37 71 90: 1 11 30 31 41 61 81: 1 22) 21 33 40 70 72 80: 1 13 32 34 42 60 90: 1 12 22 36 37 71 82: 1 11 30 31 41 61 81: 1 23) 34 36 40 42 60 81: 1 13 21 32 33 70 80: 1 12 22 37 61 71 82: 1 11 30 31 41 72 90: 1 24) 22 34 40 60 61 81: 1 13 21 32 33 70 80: 1 12 36 37 42 71 82: 1 11 30 31 41 72 90: 1 25) 34 36 40 60 71 81: 1 13 21 32 33 70 80: 1 12 22 37 41 72 82: 1 11 30 31 42 61 90: 1 26) 21 30 33 61 80 81: 1 13 34 40 60 70 71: 1 12 22 37 41 72 82: 1 11 31 32 36 42 90: 1 27) 21 30 33 61 80 81: 1 13 34 40 42 70 90: 1 12 22 36 37 71 82: 1 11 31 32 41 60 72: 1 28) 21 30 33 61 80 81: 1 13 22 34 40 70 90: 1 12 36 37 42 71 82: 1 11 31 32 41 60 72: 1 29) 21 30 33 61 80 81: 1 13 22 34 40 70 90: 1 12 36 37 42 60 72: 1 11 31 32 41 71 82: 2 30) 21 30 33 61 80 81: 1 13 34 40 42 70 90: 1 12 22 31 32 41 72: 1 11 36 37 60 71 82: 1 31) 22 31 34 41 71 72: 1 13 21 32 33 70 80: 1 12 30 36 40 61 90: 1 11 37 42 60 81 82: 1 Pentacube-Puzzle Contest 98: Torsten Sillke, 27.04.98, Frankfurt/M. The dissection is fixed. The holes for the corner blocks are fixed. The center block shall contain two 3-stack holes at the bottom. The center stack contains no holes. This is the only possible partition of this dissection with this constraints. 31 34 36 50 71 90: 1 21 30 33 61 80 81: 1 12 22 37 41 72 82: 1 11 13 32 40 42 70: 1 10 20 35 51 60: 2 The solution contains a symmetric subblock of (20, 35). Reflection of the symmetric blocks (10), (20, 35), (51), and (60) gives the other solution. 71 71 x . 61 80 x . 41 41 x . 13 42 x . . 60 . 90 50 90 x 61 80 81 x 22 41 72 x 13 42 70 x 10 60 x 90 90 90 36 80 80 80 21 22 22 22 12 13 42 11 11 . x . 71 50 x . 61 61 x . 41 82 x . 13 40 x . . 60 . 71 50 36 x 61 21 81 x 72 72 72 x 70 70 70 x 10 60 x . 50 36 36 . 21 21 21 . 72 22 12 . 42 42 11 . x . 71 34 x . 30 30 x . 41 82 x . 13 40 x . . 35 . . 50 36 x . 81 81 x . 82 82 x . 70 32 x 10 60 x . . 31 31 . . 81 33 . . 12 12 . . 32 11 . x . . 34 34 . . 30 30 . . 82 37 . . 40 40 . . 35 . . . 34 31 . . 33 33 . . 37 37 . . 32 32 10 35 35 51 . . . 31 . . . 33 . . . 12 . . . 11 . 20 . ^ | . . 34 . . . 30 . . . 37 . . . 40 . . 20 . . 20 . . . . 31 . . . 33 . . . 37 . . . 32 10 20 35 -> 51 51 51 . . . . . . . . . . . . . . . . . 20 . . 51 .