You may be familiar with Dudeney's famous dissection of the equilateral triangle into a square. (A nice physical version is demonstrated here.) His dissection uses four pieces. I believe this to be the minimum (though I have never seen a proof).
Now, I shall give you superpowers: you may dilate (aka resize, aka grow and shrink) any piece or pieces.
With this new ability, how few pieces are required to dissect the equilateral triangle into the square? (Said another way, you must partition the equilateral triangle and square each into pieces such that each piece of one is similar (but not necessarily congruent) to a corresponding piece in the other.)
This is a golf-style question: fewest pieces wins.