A hexagon can be divided into 3 pieces to make a rectangle.
Can we prove 3 pieces is minimal?
For a equilateral triangle to square dissection, it's thought that 4 pieces is minimal. We can prove that at least 3 pieces are needed by the pigeonhole principle. With just 2 pieces, one of the length $2/\sqrt[^4]{3} = 1.51967$ edges of the unit area triangle must fit on the unit area square, but that's impossible. Therefore, at least three pieces are needed for dissecting an equilateral triangle into a square.
Is there a proof that a regular hexagon cannot be cut into 2 pieces to make a rectangle?
Update: an NJA Sloane paper.