Just illustrating [Noam Elkies' 4-piece solution](http://mathoverflow.net/a/266073/6094):
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[![Elkies4Piece][1]][1]
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
[![ElkiesCube3D][2]][2]
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<sup>Bottom face is mostly yellow (except for a little green); two hidden back faces are mauve.</sup>
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  [1]: https://i.sstatic.net/CEmKc.jpg
  [2]: https://i.sstatic.net/mvhFD.jpg