Monday, March 25, 2013

Hoffman's packing puzzle

This is a photo of my Bill Cutler version of Hoffman's packing puzzle which I have had socked away in a cabinet for the better part of thirty-five years. The blocks are 15x18x22 deci-inches and the frame, 55 cubed. Bill's pieces (each one composed of a 7.5x18x22 doublet) sport some rough saw-cut ends and sides. The frame was originally a box, but one of its sides warped and I decided that it would look better with that and another one of its sides removed. Fine woodworking versions have been created by Trevor Wood and John Devost. Gemani Games and Puzzles sells a version in Samanea.

Dean Hoffman thought up the packing problem in 1978 (Bill Cutler thinks it was 1976: see #8 here) and wrote about it in David Klarner's The Mathematical Gardner (1981, pages 212-225). Elwyn Berlekamp, John Conway, and Richard Guy covered it in their Winning Ways (1982, volume 2: pages 739-740, 804-806; 2004 second edition, volume 4: pages 847-848, 913-915). Alexey Spiridonov published a very nice article about it and its solutions in 2003 and posited an approach to solving the four-dimensional analogue. I don't know if this has yet been accomplished. George Miller, on one of his Puzzle Palace pages, has Donald Knuth searching (in 2004) for solutions to a 3x4x5 version of Hoffman's problem and finding three where one could squeeze an extra 28th block into a 12-cubed frame!

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