Post #677037

2026-03-16 19:52 UTC

I just stumbled across this. I haven’t tried it yet. (It’s printing now.) I had not heard about this interesting discovery! https://www.printables.com/model/221119-knuths-packing-puzzle “In 1978 Hoffman proposed that a if you take cuboids with size A×B×C, you can always pack 27 into a cube that has edges of length A+B+C. In 2003 Knuth showed that for a special subset you can fit 28 cuboids into a cube.”

Replies (3)

@robinhouston@mathstodon.xyz 2026-03-17 13:30

I just realised that Hoffman's result is a geometric version of the 3-variable case of the AM-GM inequality. I’m sure that was very obvious to some of you! \[ \frac{a+b+c}3 \ge \sqrt[3]{abc} \] iff \[ (a+b+c)^3 \ge 27abc \] for positive real a, b, c.

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@domotorp@mathstodon.xyz 2026-03-19 17:50

@robinhouston https://www.math.toronto.edu/~drorbn/projects/ArithGeom/

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@svat@mathstodon.xyz 2026-03-19 22:32

@robinhouston All the details I was able to find, based on these: https://market.cubicdissection.com/listing/perfect-packing/120685 http://www.ageofpuzzles.com/Masters/DonaldEKnuth/DonaldEKnuth.pdf (has solutions / spoilers) https://puzzlewillbeplayed.com/Brick/Hoffman/ https://www.printables.com/model/221119-knuths-packing-puzzle https://en.wikipedia.org/wiki/Hoffman%27s_packing_puzzle In 1978, Dean G. Hoffman posed this puzzle "to a conference at Miami University": Can 27 identical cuboids of dimensions a×b×c be placed into a cube of side length a+b+c? To avoid easy solutions that occur when min(a,b,c)(a+b+c)/4. Hoffman wrote in 1981 that this puzzle typically takes "20 minutes to multiple hours" to solve, and it was first solved by David A. Klarner (his doctoral advisor). There are 21 solutions. In 2004, Donald Knuth looked into the case where min(a,b,c)=(a+b+c)/4, and discovered three solutions (easy, medium, hard) that accommodate not 27 but 28 a×b×c cuboids. This puzzle was George Miller’s exchange gift at the International Puzzle Party 25 held in Helsinki, Finland 2005.

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