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?

The closest packing density is a circle.

Consider a square of 9 x 9, put your first circle in the middle, centre at coordinates (4.5, 4.5).  Arrange around them six (touching each other) new circles. Repeat this until you hit the squarte walls.  You now have 5 circles that contain 1, 6 , 12, 24 and 48 circles or 91 circles in total

The problem spoke of 10 x 10 , so you can still lay 10 circles horizontally + 9 circles vertically on the 10 row/column.

There is still some place left, on the four corners of the 9 x 9 square, if I am correct, there is enough place for on e or more coins, buit somebody else should do the math.

Posted by Hugo on 2004-07-30 08:55:47