How many ways to make a 3x3x3 cube from the seven blocks?

C B A F  # Blocks in the above figure
G  E  D

• Answer: 339

Description

• A cube consists of 27 voxels. Each voxel has three dimension.

000 001 002  100 101 102  200 201 202
010 011 012  110 111 112  210 211 212
020 021 022  120 121 122  220 221 222

• There are 7 building blocks: A .. G. Blocks have 4 voxels except that A has 3 voxels.

• The shape of a block can be depicted with voxels.

CC ..  BB .B  AA ..  FF F.
C. C.  B. ..  A. ..  .. ..

.G. ...  .EE ...   ..D ...
GGG ...  EE. ...   DDD ...

Approach

1. Find all movable position of each block
   1. Rotatation: Subgroup of symmetric group S4
   2. Translation
2. Place blocks in order
   • Depth-first search

Rotation

Position Type

Type is preserved by rotation

• Vertex

{000, 002, 020, 022, 200, 202, 220, 222}

• Edge

{001, 010, 012, 021,
 100, 102, 120, 122,
 201, 210, 212, 221}

• Face

{011, 101, 110, 112, 121, 211}

• Center

{111}

• 8Vertex+12Edge+6Face+1Center
  8eee+12eeo+6eoo+1ooo=(2e+o)^3

Rotation Operation

• Properties
  1. Preserve Even/Odd
     (a,b,c), (2-a,b,c), (a,2-b,c), (a,b,2-c)
  2. Permutation
     (a,b,c), (a,c,b), (b,a,c), (b,c,a), (c,a,b), (c,b,a)
• Simple operation: Rotation & Flip
  1. Rotation through diagonal axis
     (a,b,c) <-> (b,c,a) <-> (c,a,b)
  2. Flip against the parallel faces
     (a,b,c) <-> (2-a,b,c)
(a,b,c) <-> (a,2-b,c)
(a,b,c) <-> (a,b,2-c)
  3. Flip against the diagonal planes
     (a,b,c) <-> (a,c,b)
(a,b,c) <-> (b,a,c)
(a,b,c) <-> (c,b,a)

Solution

#001 (0+000, 0+100, 2+111, 0+002, 3+021, 2+010, 8+200)
AAD  BBD  GBD
FAD  BCC  GGC
FFE  FEE  GEC

• A cube has the three planes of 3x3.
• Alphabets indicate the block id's.
• In i+ddd, ddd represents the translation in each axis.

Block in string format

CC.  ...  ...
C..  C..  ...
...  ...  ...

has the bounding box of size (2,2,2):

11  00
10  10

where 1 is the occupied position by the block and 0 is the empty position. The sequence is serialized as 11 10 00 10 in the tensor order.
The dimension of the bounding box 222 and the occupancy sequence 11100010 are combined into '22211100010'.