185 geometry/hypercube.p
Description
This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.
185 geometry/hypercube.p
How many vertices, edges, faces, etc. does a hypercube have?
geometry/hypercube.s
Take any vertex of the hypercube, and ask how many k-V's it
participates in. To make a k-V it needs to combine with k adjacent and
orthogonal vertices, and there are (nCk) distinct ways of doing this
[that is, choose k directions out of n possible ones]. Then multiply
by 2^n, the total number of vertices. But this involves multiple
counting, since each k-V is shared by 2^k vertices. So divide by 2^k,
and this yields the answer: (nCk)*2^{n-k}.
For example, 12d hypercube:
0-v: 4,096 1-v: 24,576 2-v: 67,584 3-v: 112,640 4-v: 126,720 5-v: 101,376 6-v: 59,136 7-v: 25,344 8-v: 7,920 9-v: 1,760 10-v: 264 11-v: 24 12-v: 1
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