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One in a Trillion

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

A B C D E F G H

Pieces	Selection	Goal	Holes	Solutions
8	random	4×4×4	0	1
		9×9×1	17	1

Darrah Flat

Bill Darrah defined a class of polycube pieces.

• connected
• snug in 4×4×1
• contain no 2×2 part

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.

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 Set	4×4×4	9×8
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	0	One 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	1	Impossible 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 Set	4×4×4	9×8
1.	79×4 144×4	2/3	0
2.	79×2 90×2 144×2 224×2	2/2	1	One 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