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 started doing the calculations for the problem, when I noticed a serious error in my former posting (the circular squeezing technique I proposed). 

The inner circle that in a 9 x 9 grid would have its centre on coordinates (4.5, 4.5) has indeed 6 circles that can be arranged around it.  Around that 2ndcircle, you can arrange 12 more circles.  But the 3rdcircle has 18 (and not 24) circles, the 4thcircle has 24 circles (not 48).  This gives a total of 61 circles.

If you let rest the circle construction on the bottom, then it is easy to see that the height of our fourth circle is 9 units and you can calculate that the width is 8 units, so we can put two more vertical columns, giving a total of 81. Put one column to the far left, one to the far right. 

In our circle construction the top circle at the most left column (Column 2) is at the same height as the top circle of the 3rdcircle (The one with 12 circles) such that there is place above for 3 more circles.  The bottom circle has room for two 2 circles.  This is the same at the right side.  So now we are at 81 +2 x 3 = 87

When going more to the centre you can find that for the following three columns there is room for 2, 1, 1 circles at the bottom and 2,2, 1 at the top.  The whole thing is the same at the right side.  So we have 2 x (2+1+1+2+2+1) extra circles, giving 87+18 = 105

At the center column, above the outer circle construction, you can put one more circle.

This gives me a total of 106 circles.

Posted by Hugo on 2004-08-02 15:04:14