TAOCP 7.2.2.1 Exercise 356

Exercise 356. [27] [27] Polysphere puzzles often involve the construction of three kinds of shapes:

n-tetrahedron $\qquad$ $m \times n$ roof $\qquad$ stretched

(as seen from $\qquad$ (shown for $\qquad$ $m \times n$ roof

the top, for $\qquad$ $m = 3$, $\qquad$ (as seen from

$n = 4$) $\qquad$ $n = 4$) $\qquad$ $n{=}3, n{=}4$)

(An $n \times n$ roof or stretched roof is called an "$n$-pyramid" or a "stretched $n$-pyramid".)

a) Define each of these configurations by specifying a suitable base placement.

b) Each of the shapes mentioned can be made of tetraspheres, and so is the stretched $4 \times 3$ roof. Find all multisets of five tetraspheres that suffice to make these shapes.

c) The 4-pyramid and the stretched 4-pyramid involve 30 spheres. What multisets of ten trispherex are able to make them?

d) The truncated octahedron, which represents all permutations of ${1, 2, 3, 4}$, is a noteworthy 24-cell subset of $S$ (see exercise 5.1.1–10). What multisets of six tetraspherex can build it?

Verified: no
Solve time: 5m41s

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