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21 - Cubes Rationalized

Published online by Cambridge University Press:  05 August 2012

Greg N. Frederickson
Affiliation:
Purdue University, Indiana
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Summary

In the twilight of his career at Cambridge University, Herbert W. Richmond combined his lifelong interests in geometry and number theory to pose one of the more natural of puzzles in solid dissections: Dissect cubes of edge lengths 3, 4, and 5 to give a cube of edge length 6. The venerable octogenarian produced a dissection with some appealing symmetries, but it used twelve pieces – a rather generous number.

Six years later, John Leech, then a student at Cambridge, latched onto the problem. He challenged the readers of Eureka, the Cambridge undergraduate math journal, to find a dissection that used at most ten pieces, an improvement of two over Richmond's solution. And within a year, the charge of that young generation was complete. Roger Wheeler, also an undergraduate at Cambridge, responded with a dissection that reduced the number of pieces even further, again by two, down to eight!

What colossal impertinence in that young generation at Cambridge! Or was it just a colossal impatience, given the few solid dissection problems that had been posed and solved? Surely it was implacability in the face of problems that seemed so much more difficult than their two-dimensional analogues. It had been a natural step for Richmond (1943) to pose the dissection for cubes realizing 33 + 43 + 53 = 63. And considering that there were no previous dissections for any solution to x3 + y3 + z3 = w3, twelve pieces might have seemed a veritable bargain.

Type
Chapter
Information
Dissections
Plane and Fancy
, pp. 247 - 257
Publisher: Cambridge University Press
Print publication year: 1997

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