Table 1.
A sample of enumerative results for the building game [36]. All results were determined by exhaustive enumeration with a computer; we are not aware of theoretical bounds or an exact enumeration formula for the building game. The numbers in brackets refer to shellable intermediates, edges between shellable intermediates, and pathways through shellable intermediates, respectively (see [80] for more on shellability). The number of intermediates for the Platonic solids is 1 more than in the results from [23], since we also count the complete polyhedron as an intermediate.
| Polyhedron | No. of faces | No. of intermediates | No. of edges in  | No. of assembly pathways from □ to ▪ |
|---|---|---|---|---|
| Tetrahedron | 4 | 5 (5) | 4 (4) | 1 (1) |
| Cube | 6 | 9 (8) | 10 (8) | 3 (2) |
| Octahedron | 8 | 15 (12) | 22 (14) | 14 (4) |
| Dodecahedron | 12 | 74 (53) | 264 (156) | 17,696 (2,166) |
| Icosahedron | 20 | 2650 (468) | 17242 (1984) | 57,396,146,640 (10,599,738) |
| Truncated tetrahedron | 8 | 29 (22) | 65 (42) | 402 (171) |
| Cuboctahedron | 14 | 341 (137) | 1636 (470) | 10,170,968 (6,258) |
| Truncated cube | 14 | 500 (248) | 2731 (1002) | 101,443,338 (5,232,294) |
| Truncated octahedron | 14 | 556 (343) | 3071 (1466) | 68,106,377 (5,704,138) |
| Polyhedron | No. of faces | No. of intermediates | No. of edges in | No. of assembly pathways from □ to ▪ |
|---|---|---|---|---|
| Tetrahedron | 4 | 5 (5) | 4 (4) | 1 (1) |
| Cube | 6 | 9 (8) | 10 (8) | 3 (2) |
| Octahedron | 8 | 15 (12) | 22 (14) | 14 (4) |
| Dodecahedron | 12 | 74 (53) | 264 (156) | 17,696 (2,166) |
| Icosahedron | 20 | 2650 (468) | 17242 (1984) | 57,396,146,640 (10,599,738) |
| Truncated tetrahedron | 8 | 29 (22) | 65 (42) | 402 (171) |
| Cuboctahedron | 14 | 341 (137) | 1636 (470) | 10,170,968 (6,258) |
| Truncated cube | 14 | 500 (248) | 2731 (1002) | 101,443,338 (5,232,294) |
| Truncated octahedron | 14 | 556 (343) | 3071 (1466) | 68,106,377 (5,704,138) |