Coins of diameter 1 have been placed in a square of side 11, without overlapping or protruding from the square. Can there be 126 coins? and 127? and 128?
First off, I found it easier to work with radius 1 circles and then have the square of side 22. This scaling doesn't change anything.
A rectangular packing can easily fit 11*11=121 circles. It wastes space between circles.
A hexagonal packing is much more dense and can easily fit alternating rows of 11 and 10 and six of each for a total of 126.
https://www.desmos.com/calculator/3hd74pyl9m
This also seems to waste space the the edges of the row of 10 and also at the top.
Erich Friedman's packing center only goes up to 24 circles, but hints that a combination of rectangular and hexagonal would likely help mitigate the issues of wasted space.
https://erich-friedman.github.io/packing/cirinsqu/
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Posted by Jer
on 2025-04-23 08:10:01 |
A start: 126
