Soma cube TB#3: The T Piece

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The Soma cube is a solid dissection puzzle invented by Piet Hein. Seven pieces made out of unit cubes must be assembled into a 3×3×3 cube. The pieces of the Soma cube consist of all possible combinations of three or four unit cubes, joined at their faces, such that at least one inside corner is formed. There is one combination of three cubes that satisfies this condition, and six combinations of four cubes that satisfy this condition, of which two are mirror images of each other. Thus, 3 + (6 × 4) is 27, which is exactly the number of cells in a 3×3×3 cube. There are 240 distinct solutions of the Soma cube puzzle.

Piece #3, or T, is made up of 4 unit cubes, shaped like an "T". The "T" is special mathematically- in all 240 distinct solutions, the T always appears in the same orientation. Each solved cube can be rotated such that the "T" piece is on the bottom with its long edge along the front and the "tongue" of the "T" in the bottom center cube. This can be proven as follows: If you consider all the possible ways that the "T" piece can be placed in the large cube (without regard to any of the other pieces), it will be seen that it will always fill either two corners of the large cube or zero corners. There is no way to orient the "T" piece such that it fills only one corner of the large cube. The "L" piece can be oriented such that it fills two corners, or one corner, or zero corners. Each of the other five pieces have no orientation that fills two corners; they can fill either one corner or zero corners. Therefore, if you exclude the "T" piece, the maximum number of corners that can be filled by the remaining six pieces is only seven (one corner each for five pieces, plus two corners for the "L" piece). A cube has eight corners. The "T" piece cannot be oriented to fill just that one remaining corner, and orienting it such that it fills zero corners will obviously not make a cube. Therefore, the "T" must always fill two corners, and there is only one orientation in which it does that. The "T" is therefore considered the normalized piece of a Soma cube.