(spoiler below for the first question) Jim Propp wrote: ----- Puzzle: Show that a triangle cannot be written as the union of a finite number of parallelograms with disjoint interiors. Puzzle: Show that you can cover n/(n+1) of the area of a triangle using n parallelograms with disjoint interiors. Problem: is it possible to do better? ----- Nice question! (SPOILER ALERT) ----- Starting from one side of the triangle, each parallelogram must share a side with a new adjacent parallelogram having a new parallel side. This cannot be one of the triangle sides, since those cannot be parallel, contradicting the finiteness of the set of parallelograms. ----- (Incidentally, with countably infinitely many parallelograms, it's possible to cover the interior and two sides of a triangle, but I'm not sure if the entire third side can be covered; I suspect not.) —Dan