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?

i believe that you can fit 105 circles intact inside the 10 x 10 space.

picture four circles in a 2 x 2 space. take away one of the top circles and let the other fall into the space between the bottom two. this has a height of 1 + sqrt(3)/2.

now picture the 10 x 10. number the rows 1-10 with row 1 on bottom. pull one circle from each of the even numbered rows and let them fall into their closest packing.

(row 1 has ten circles and row 2 has nine, etc.)

you now have 95 circles. the height is now 9 x sqrt(3)/2 + 1. which is, according to my calculator, 8.794.

that leaves room on top for one more row of ten, which gives us 105 circles.

Posted by rixar on 2004-07-30 20:12:41