Colonel's Bouquet

This picture shows four hexacubes—solids formed by joining six identical cubes:

Join any pair of pieces to make a symmetrical shape.

Finally, join all four pieces to make a solid rectangular box.

This was Nigel Croot's exchange puzzle at the 2017 Nob Yoshigahara Puzzle Design Competition. Nigel also commissioned Brian Menold of Wood Wonders to make copies in exotic wood for the judging. On occasion Brian makes copies for sale.

I am indebted to László Molnár for suggesting that I investigate symmetric pairs of hexacubes, based on Péter Gál's work on symmetric pairs of pentacubes. Puzzle aficionados may detect a resemblance between Colonel's Bouquet and Péter's Symptomino Puzzle.

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