A favorite fun problem I give to students and (even non-mathematical) friends is the following:
Find a decomposition of an equilateral triangle into three similar polygons, exactly two of which are congruent.
There is a cute solution which the persistent reader will find. (By the way, the problem is not mine. I believe I saw it in an old Hungarian problem book, but I don't recall for sure.)
The funny thing is that everyone seems to find the same solution as my own. I sort of expected other solutions to crop up. So I am led to ask:
Is there a unique such decomposition (up to isometry)?
Apologies if this pushes the MO boundaries a little. I am probably not alone in wondering about this, though. And it seems to compare well with a few other "elementary geometry" questions on MO.
For a hint leading to (what I think is) the standard (unique?) solution, see this: http://en.wikipedia.org/wiki/File:Tile_3,6.svg