2

One in a Trillion


Design and Copyright : Joseph Becker (2001). IPP22. Puzzlecraft.

A B C D
E F G H
PiecesSelectionGoalHolesSolutions
8random 4×4×401
9×9×11719×9
thankful to Antoine Bostsarron

Darrah Flat

Bill Darrah defined a class of polycube pieces.

Joseph Becker calls this "Darrah Flat", and analyzed this.

Heptacubes : 18 pieces
Octocubes : 47 pieces
Nonacubes : 84 pieces
Decacubes : 80 pieces
Undecacubes : 45 pieces
Dodecacubes : 12 pieces

Darrah Flat Octocubes are as follows.

668/66 698/69 768/76 798/79 908/90 918/91 938/93 948/94 958/95 968/96 1448/144 1468/146 1518/151 1528/152 1538/153 1758/175 1768/176 1778/177 1788/178 1808/180 1948/194 1958/195 1978/197 1988/198 2038/203 2048/204 2058/205 2068/206 2108/210 2248/224 2258/225 2628/262 2658/265 2668/266 2678/267 2698/269 2708/270 2848/284 2878/287 2888/288 2898/289 2908/290 2918/291 3438/343 3448/344 3458/345 3468/346

There are 47C8 = 314,457,495 sets of 8 different Darrah Flat Octocubes, and there are 17 sets can form a 4×4×4 cube, but 5 sets are unassemblable.

 Piece Set4×4×49×8 
1.69 91 144 194 204 210 270 291 1/1 0
2.69 95 194 197 204 210 269 291 1/1 0
3.69 96 178 194 197 204 210 291 1/1 0
4.69 96 178 194 197 205 210 290 1/1 0
5.90 144 176 177 206 225 267 291 1/1 0One in a Trillion
6.90 146 151 176 177 195 267 290 0/1 10
7.90 146 151 176 177 224 267 343 0/1 0
8.90 153 176 225 266 267 287 291 1/1 0
9.90 175 177 180 225 265 266 287 0/1 0
10.90 177 178 224 225 266 291 345 1/1 0
11.90 177 210 224 225 265 288 291 1/1 0
12.91 93 151 194 210 224 270 291 1/1 0
13.91 144 194 206 210 225 270 291 1/1 0
14.91 146 151 152 262 265 287 288 0/1 1Impossible One in a Trillion
15.91 146 151 176 177 195 267 270 0/1 10
16.94 96 151 178 206 210 224 284 1/1 0
17.94 96 152 176 206 210 224 284 1/1 0

When pair pieces are used, there are 7 sets, but 3 sets are unassemblable.

 Piece Set4×4×49×8 
1.79×4 144×4 2/3 0
2.79×2 90×2 144×2 224×2 2/2 1One in a Trillion Lattice
3.79×2 90×2 198×2 262×2 1/1 0
4.90×4 224×4 3/3 0
5.90×4 266×4 0/4 0
6.146×4 225×4 0/8 0
7.146×4 225×2 270×2 0/2 1

Dec 8, 2002 by k16@chiba.email.ne.jp