# Face-Matching Polycubes

2015/09/03

After I had explored some polyominoes with notches at their edges Jared McComb wanted to go 3-D and suggested putting pegs and notches at the whole surface of a polycube. The whole surface of a dicube consists of 10 squares and with the Polya-Bernstein formula we get 1/8*(10*9+3*2+0+0+0+0) = 12 different pieces with exactly one peg and one notch. Due to a checkering argument these pieces don't allow for a box with a flat surface where all pegs match the notches. Therefore I added the 2 dicubes with only one peg and the 2 dicubes with only one notch. The picture above shows the set and it's possible to construct boxes and enlarged tetracubes with it.

If you can't get a solution the layers of a construction are provided. Black and white triangles show the pegs and notches, whereas colored triangles indicate touching cubes of the same piece in the layer above or below.

In the table below some sets and constructions are listed. Click the different polycubes for more information. The used pegs or male square faces and the used notches or female square faces are indicated by m and f, respectively.

 m: one male square face f: one female square face mf: one male and one female square face mm: two male square faces ff: two female square faces

Type of Square Faces Number of Pieces Total Volume Constructions
Dicube mf 12 24 No boxes
mf or f or m 16 32 2x2x8 and 2x4x4 boxes; enlarged tetracubes
mm or ff 9+9 36 3x3x4 and 2x3x6 boxes
I-Tricube mf 24 72 2x3x12, 2x4x9, 2x6x6, 3x3x8, 3x4x6 boxes
f or m 3+3 18 No 2x3x3 box
ff or mm 16+16 96 Two 2x3x8 boxes
V-Tricube mf 91 273 No boxes, but many boxes and decomino models with one piece left
f or m 7+7 42 2x3x7 box
mf or m or f 105 315 3x5x21, 3x7x15, 3x3x35, 5x7x9 boxes
mm or ff 49+49 294 3x7x14 box
I-Tetracube mf 39 156 3x4x13 box
mm or ff 25+25 200 2x4x25, 2x5x20, 4x5x10, 5x5x8 boxes
Solid Square Tetromino mf 30 120 2x3x20, 2x4x15, 2x5x12, 2x6x10, 3x4x10, 4x5x6 boxes
m or f 2+2 16 2x2x4 box
mm or ff 20+20 160 Two 2x4x10, two 2x5x8 boxes
Solid L-Tetromino mf 306 1224 4x6x51 box
m or f 18+18 144 Two 3x4x6 boxes, two 2x3x12 boxes

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