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?

So far most answers have considered "vertical squeezing".  One answer discussed starting in the center and making concentric circles, call it "circular squeezing".

What if you rotate the square 45 degrees, put one coin in the bottom corner, then a row of 2 on top, then 3:  "diagonal squeezing" ?  I haven't done the math on this but it seems to me this might pack tighter than vertical squeezing.  More degrees of freedom.

Of course then there's a trick answer, dare I say "thinking outside the box" (pun intended, sorry)  of putting the coins on edge.  Then depending on how thick a penny is compared to its diameter you might get 500 or 600.  But I'm sure this is not the intended answer.   On edge isn't in fact overlapping.

Posted by Larry on 2004-07-31 23:33:03