Comes up with this puzzle idea while watching the little kids play. Warning first of all - I don't have an answer for it so it might be a joint effort to find out the best answer. Here we go:
1) a 50 cm x 50 cm board
2) Regular Polygons starting from triangle with only 1 of each kind (1 square, 1 pentagon, etc) and no circle. All sides of each polygon are 1 cm long.
You're to fit the max number of polygons onto this board. Here are the rules:
1) You cannot place any polygon outside the edge of the board.
2) All polygons are placed flat (i.e. you cannot 'standing up a triangle to save spaces) and cannot overlaps.
3) Each polygon can only have number_of_side - 2 side(s) touching other polygon. I.e. a triangle can only have 1 side touching other polygon while a octagon can have 6.
4) For simplicity purpose, we can ignore spaces between 2 edges (If we put two '1cm x 1cm square' side by side for an example, it results in 2 cm long rectangle)
5) Again for simplicity purpose (which is a good candidate for next level), any touches between 2 sides will be counted as one touch:
Corners touching corners / side won't be counted.
Suggestions to refine the question are welcome.