At a workshop it was suggested that it likely remains an open problem whether or not there is a 3- or 2-piece dissection of a square to an equilateral triangle. Can anyone confirm that this is unresolved?

Four-piece dissections are known, the most famous being Henry Dudeney's century-old gem:
                  http://cs.smith.edu/~orourke/MathOverflow/Dudeney.gif
_{[Maple image from this link.]}