[math-fun] Packing a 2-by-n rectangle with disks
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, Jim
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,
Jim _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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,
Jim _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Yes, more details please! I'll be featuring the problem in my next blog post a week from now, and since it'll be read by a few hundred people, I'd like to give credit where credit is due. Thanks, Jim On Wednesday, March 9, 2016, Nick Baxter <nickb@baxterweb.com> wrote:
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,
Jim _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Hi Folks, Regarding the problem of dense packings of unit discs in a 2 by n rectangle, here is the history as far as know. The problem was first posed by Fejes-Toth many years ago, at least by 1971. (see the attached 1991 article by Furedi). Actually, the problem was to determine the density of the densest packings in an infinite rectangle of width 2. Fejes-Toth had a conjecture which I showed was wrong and conjectured that the "alternating triangles" packing was optimal (in 1986). Furedi proved that this was correct and that his result worked all the way up to rectangles of width 1+sqrt(3). Of course, once the density is greater than 2, then by choosing n large enough, you can get more than 2n discs into a 2 by n rectangle. I showed some pictures of this in a book I published (in Japanese!) with Jin Akiyama in 1993 (attached) where I mentioned that n = 1000 was certainly enough to squeeze in a few extra discs. I don't know what the smallest n is which allows you to pack 2n+1 discs in a 2 by n rectangle. Certainly it is more than 100. Best regards, Ron On Wed, Mar 9, 2016 at 8:29 AM, rkg <rkg@ucalgary.ca> 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,
Jim _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (4)
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James Propp -
Nick Baxter -
rkg -
ronald graham