I recall first seeing this from Nob Yoshigahara. Dick Hess and I studied it years ago; I'll have to dig up my notes; we may have improved the previously known packing, but not n. And Dick may have included it in one of his recent books math puzzle books. You can contact me privately for more. Nick On 3/9/2016 8:29 AM, rkg wrote:
Mainly to Jim Propp, The people who can help you as well as anyone are Elwyn Berlekamp and, particularly, Ron Graham. I'm copying this to them in the hope that they'll be forthcoming. There's a session on the Saturday afternoon of MathFest that might be an appropriate place to mention this problem. See you (all!) there. R.
On Wed, 9 Mar 2016, James Propp wrote:
I'd like a mini-bibliography on the problem of packings disks of diameter 1 in a 2-by-n rectangle. So far all I've found is problem 211 in "The Inquisitive Problem Solver" (by Vanderlind, Guy, and Larson --- hi, Richard); those authors credit Eugene Luks for passing along the problem, but they don't say who first noticed/proved that you can fit more than 2n disks of unit diameter into a 2-by-n rectangle.
In addition to seeking the original source for this paradox, I'd like to know of places that write about it clearly. (Is it somewhere in Gardner's works?)
Also, the solution provided by Vanderlind et al. isn't the optimal packing; as was recently discussed in this forum (Bill Gosper being the discussant who comes to mind), one can do better with a pattern that repeats every 6 disks. Is that pattern published somewhere?
For that matter, where is that picture? Someone posted a link to it, but Google won't help me find it.
Thanks,
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