# the Soma cube

The Danish Piet Hein called Soma cube this puzzle whose seven pieces are convex assemblages of at most four identical elementary cubes assembled by their faces.
 S1 S1 is the only assembling of three elementary cubes. S5 and S6 are mirror images: two S5 (or two S6) can be assembled to build a cube, this is impossible with one S5 and one S6! S2 S3 S4 S5   S6 S7

The seven pieces assemble themselves in a big cube, the Soma cube, constituted of 27 elementary cubes.
Here is one of the 240 different ways to put the seven Soma pieces together
(rotations and reflections don't give different arrangements).
John Conway proposed a (non unique) notation to describe these assemblages: on three 3x3 grids representing the three layers of elementary cubes, the integers n (from 1 to 7) show the position of an element of a piece Sn.
 top 2   4   3 5   4   4 5   5   4 middle 2   6   3 5   6   3 1   1   7 bottom 2   2   3 6   6   7 1   7   7

The Soma pieces and cube lead to numerous problems; the following two are really challenging:
• S7 can have four different positions in the cube (above, on a vertex); find assemblages showing the three others cases,
• build S1 at the scale 2 (assemblage of 3x8=24 elementary cubes) using the six other pieces S2 to S7.

 references: •  Polyèdres et autres bidules de l'espace ordinaire, Géométrie expérimentale au collège, CRDP Versailles, 1995 (in French) • SOMA by Thorleif

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