Into how many regions can you partition the plane with m n-sided regular polygons?

For example, with two squares you can achieve up to 10 regions by choosing the right size and position of your squares.

Although I don't think this puzzle has been completely solved just yet, I would like to propose the following generalization:

Into how many regions can you partition the plane with m n-sided simple polygons?

Investigating this more general problem may be helpful in understanding how the rather special situation of regular polygons can be treated in order to formally deduct a formula. For the general  problem I have a solution for even n. For odd n the problem seems to be trickier. Anyone wants to give it a try?

Posted by JLo on 2006-09-09 07:20:52