At a workshop it was suggested that it likely remains an open problem
whether or not there is a 3- <strike>or 2</strike> -piece 
<a href="http://mathworld.wolfram.com/Dissection.html">dissection</a>
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:
<br />
<img src="http://cs.smith.edu/~orourke/MathOverflow/Dudeney.gif" />
<img src="http://cs.smith.edu/~orourke/MathOverflow/DissectionFixed.gif" />
<br />
<sub>
[Maple image from this
<a href="http://www.maplesoft.com/applications/view.aspx?SID=6499">link</a>.]
</sub>
<br />