What is the minimum of squares to fit within rectangular box 10 x 16?

Question

I'm trying to apply circle packing data to a 10 x 16 inch sheet for printing, here: http://hydra.nat.uni-magdeburg.de/packing/csq/csq.html#Applications

And I want to achieve the least waste by figuring out if I can divide this 10x16 rectangle into squares, the minium of squares beyond the waste from one 10 x 10 square, and calculate the ideal amount of circles.

Basically I'm trying to figure out a rough, yet fairly accurate way to calculate how many designs of given sizes can fit on one 10 x 16 page with minimal waste. I would assume that circles generate less waste than squares, but maybe given the waste of converting it into squares first, squares would be a better way to estimate instead?

This is really quite confusing. If you are trying to pack squares into a rectangle, why do you mention circles? — Chris Godsil, Jan 31 '13 at 01:21
No $16 \times 16$ squares fit in a $10 \times 16$ rectangle, so that would seem to be the minimum. — Ross Millikan, Jan 31 '13 at 01:27
A more general question was asked here Depending on the size of the circle, it will be more efficient to pack them hexagonally than square pack. — Ross Millikan, Jan 31 '13 at 01:30

score 1 · Answer 1 · answered Jan 31 '13 at 01:36

1

You can do $10+6+4+2+2$ for $5$ squares. I doubt it can be beat.

answered Jan 31 '13 at 01:36

Ross Millikan

383,099

OEIS sequence A219158 confirms that this is optimal. – RavenclawPrefect Feb 01 '21 at 17:44

What is the minimum of squares to fit within rectangular box 10 x 16?

1 Answers1