David, I'd very much like to see that. Do you have Mathematica 6 yet? if not, I might be able to arrange something for you. --Ed Pegg Jr "David W. Cantrell" <DWCantrell@sigmaxi.net> wrote: If anyone finds improved packings of unit circles in ellipses, I would greatly appreciate their sending me a copy of whatever they send Erich. Furthermore, for anyone wanting to work on that packing problem, I can easily provide, on request, details of all the packings in the form of a Mathematica (v. 5.2) notebook. David W. Cantrell ----- Original Message ----- From: "James Buddenhagen" To: "math-fun" Sent: Sunday, October 07, 2007 14:51 Subject: [math-fun] packing circles

Bill Gosper's packing circles in an oval puzzle reminded me of the following packing problem:

Find the ellipse of smallest area which can contain n non-overlapping unit disks.

This problem is interesting in that even for small n the answer can be non-intuitive. For example, for n = 3, I would have guessed that the best ellipse would be a circle. But it is not.

For the best known results up to n = 24 see this page http://www.stetson.edu/~efriedma/cirinel/ at Erich Friedman's packing center. If you find any improvements let Erich know and he will update the page.

Jim Buddenhagen

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Sorry, Ed, I don't know the puzzle you mentioned. But here's another one, for anyone who might be interested: How many boxes, each measuring 7 inches x 11 inches x 13 inches, can be packed into a crate measuring 4 feet x 5 feet x 6 feet? David ----- Original Message ----- From: "Ed Pegg Jr" <ed@mathpuzzle.com> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Monday, October 22, 2007 16:23 Subject: [math-fun] History of packing 27 cuboids into a cube.

I have a puzzle by Tom Lensch. 27 AxBxC cuboids, to pack into a cube of side A+B+C. I know I've seen this puzzle written up somewhere, but I can't remember where, now.

I want to add the history to a interactive demonstration I made for the problem: http://demonstrations.wolfram.com/BoxPacking/

--Ed Pegg Jr

Ed: The puzzle is described (and pictured) in The Mathematical Gardner, edited by David Klarner, in an article by D. G. Hoffman called Packing Problems and Inequalities. There's another, older, article by de Brujin (sp?), I think, but I'll have to find it. According to George Miller (see below) Hoffman posed the original packing problem at a conference at Miami university in 1978. At any rate, in the original problem the cuboids just need to satisfy an inequality (sindes a, b, c must be different and the smalles dimension must be larger than (a + b + c) / 4 . Don Knuth asked the question of what happens with equality, and found there are actually 3 different ways to pack 28 cuboids in cube. George Miller made a version of this, called Perfect Packing. He says it will be on his Puzzlepalace.com website, when he gets to it. -- Stan

I have a puzzle by Tom Lensch. 27 AxBxC cuboids, to pack into a cube of side A+B+C. I know I've seen this puzzle written up somewhere, but I can't remember where, now.

I want to add the history to a interactive demonstration I made for the problem: http://demonstrations.wolfram.com/BoxPacking/

--Ed Pegg Jr

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