Ron Graham admits to knowing more about this problem, but hasn't vouchsafed anything yet. R. On Mon, 22 Feb 2016, Scott Huddleston wrote:
Thanks for the pointer, Richard. Locally optimizing two pennies at the end improves 477 pennies in 2x238 to 473 pennies in 2x236 I don't see any further improvements.
The same idea lets you pack 400 pennies in 1.964 x 200
For Gosper's original 2x200 teaser, I'd be very surprised if 401 were possible.
Scott
On Fri, Feb 19, 2016 at 10:14 AM, rkg <rkg@ucalgary.ca> wrote:
We used this as Problem 211 in The Inquisitive Problem Solver. I don't know where we got it from. Elwyn Berlekamp and Joe Buhler may have used it in their column in Emissary, the journal of MSRI. We only gave a packing of 477 balls in a 1 x 2 x 238 box. I know that better results are known to Ron Graham and to Elwyn Berlekamp. Perhaps either or both of them will tell us what they know, and how much has been proved. R.
On Fri, 19 Feb 2016, Erich Friedman wrote:
I'm a tad suspicious: are the circle packings on Friedman's page proved
to be maximal, or are they merely the best known so far? There is no link to any proofs, or key to diameters ... WFL
I had the very same question. But on some related pages, some
packings or coverings are referred to as best known and others are said to be proven the best . . . so I'm guessing that unless the word "proved" is used, they are only the best known.
It's not easy to find the absolute optimum of such things with proof.
this is correct. in general on my packing pages "Trivial" means there is an easy proof not attributed to a given person, "Proved by ..." means it has been proven optimal, and "Found by ..." means it is only the best currently known, with no proof known.
much fame awaits those who could find such proofs, but they do tend to be hard. for example, it was only in this century that the best packing of 6 unit squares in a square was confirmed to be a square of side 3. (i proved the cases for 7, 8, 14, and 15 squares a few years earlier.) i suspect 27 unit squares in a square will be the next to be proved.
http://www2.stetson.edu/~efriedma/squinsqu/s27.gif
One thing a torus-lover like me would like to see added if Erich
decides to do so is additional best-known or proven optima for the two most symmetrical tori:
when i retire in a few years, my guess is that you will see exactly that. :)
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