I could quickly prove that 3 points can always be covered. I could not immediately prove the same for 4, though I have no doubt that it's true. My intuition is that an uncoverable set can be constructed with on the order of 15 to 20 points, but I really have no idea how to go about it. On Fri, Dec 10, 2010 at 9:21 AM, Fred lunnon <fred.lunnon@gmail.com> wrote:
On 12/9/10, Ed Pegg Jr <ed@mathpuzzle.com> wrote:
What is the minimum number of points that cannot be covered by unit diameter coins? This is from Naoki Inaba ( http://www.janko.at/Raetsel/Naoki/ )
A dangerous individual, judging by this specimen ...
I suspect a fiendish oriental plot to destroy Western economies by distracting mathematicians from shopping for useless Christmas presents. WFL
A fuller explanation:
Coins (of a unit diameter) can be packed in a square or hexagonal lattice. Either packing has holes.
http://mathworld.wolfram.com/CirclePacking.html
If there were 3 points on the plane, the coins could be moved to cover the points.
If there were 3 million points on the plane, about 10% of them would be in the holes.
(Packing density is pi sqrt(3)/6)
Problem -- What is the minimum number of points that cannot be covered by coins?
--Ed Pegg Jr
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun