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.

PolyhedronNo. of facesNo. of intermediatesNo. of edges in No. of assembly pathways from □ to ▪
Tetrahedron 5 (5) 4 (4) 1 (1)
Cube 9 (8) 10 (8) 3 (2)
Octahedron 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 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)
PolyhedronNo. of facesNo. of intermediatesNo. of edges in No. of assembly pathways from □ to ▪
Tetrahedron 5 (5) 4 (4) 1 (1)
Cube 9 (8) 10 (8) 3 (2)
Octahedron 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 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)
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