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

I have taken this a step further.  I haven't decided on any conclusions at this point.

I have taken 2 hexagons and overlaid them in much the same manner as for the two quadrilaterals.

They each have 3 vertices draw to one extremity of the shape.

The complete array looks like two three-legged octopuses with their limbs across those of the other.

This arrangement has 37 enclosed regions.

I sense that having half of the vertices at one extreme of each polygon, and not being within the bounds of any other polygon, will yield the maximum number of regions.

Let me qualify the above remark, that consideration is for polygons having an even number of sides (or vertices).

Odd sets pose a different problem. [n-1]/2 vertices need to be 'at the head of the octopus' while the same number need to be the extremeties of the legs.  But where should the remaining one, that which completes the odd number be placed, in the head, or as part of the leg? 

Edited on September 23, 2006, 11:28 pm

Posted by brianjn on 2006-09-12 01:12:07