A 10x10 square can obviously hold 100 unit circles (diameter=1) when arranged in rows and columns. What is the maximum number of non-overlapping unit circles a 10x10 square can hold if the circles are packed closer together?

By symmetry, it shouldn't matter if we approach this on a row or a column basis.  So, let's consider placing the first circle in the upper left, and in fact let it continue to the right... so we get a row of ten circles along the top of the square.

Then we place a row of nine circles in the "spaces" between the ten circle row.

Then we place a row of ten circles up against that row.  We've now "lost" a circle in the 2nd row, but the third row is higher than it was in the orginal configuration.

Let's calculate how much higher it is.  Before, the centers between circles in the odd rows were exactly 2 units apart.  But now, we can use Pythagorean theorem to calculate the differences and we get (2x)² + x² = 1², where 4x is the distance we're looking for.

This leads to 4x = 4*√5/5 ≈ 1.788854 (which is less than 2 units).

So... if we write down the vertical coordinate of the center of the circles in each ODD NUMBERED row... we find we can get an 11th row:

row   vertical coordinate

 1    0.5
 3    2.288854
 5    4.077709
 7    5.866563
 9    7.655418
11    9.444272

So, we have 6 rows of 10 circles (60 circles), and 5 rows between them of 9 circles each (45 circles)...

for a total of 105 circles.

Edited on July 29, 2004, 3:14 pm

Posted by Thalamus on 2004-07-29 13:15:38