# Drilled Polycubes

2007/06/17

Let's take a polycube and drill some holes through the center of the cubes paralel to the edges. Depending on the number of holes you get different sets of distinct pieces and you can try to make some constructions like boxes or replicas.
The main objective is to get the holes matched i.e. no hole has a dead end and you can look through the construction. I found a cube of size 3 made this way from L-Trominoes at ebay, but the used set wasn't a complete set. Therefore I generated all pieces with a fixed number of holes from special polycubes and looked for possible constructions.
The following table shows the numbers of pieces and some links to constructions.

Holes 0 1 2 3 4 5 6 7 8 9 10 11 12 Sum
Tricubes
1 3 8 12 12 8 3 1 - - - - - 48
1 4 12 19 19 12 4 1 - - - - - 72
2 7 20 31 31 20 7 2 - - - - - 120
Tetracubes
1 2 6 10 13 10 6 2 1 - - - - 51
1 9 36 84 126 126 84 36 9 1 - - - 512
Pentacubes
1 10 45 120 210 252 210 120 45 10 1 - - 1024
1 10 45 120 210 252 210 120 45 10 1 - - 1024
Hexacubes
1 5 20 52 99 135 99 52 20 5 1 - - 624
1 12 66 220 495 792 924 792 495 220 66 12 1 4096

Let n be the maximum number of holes for a certain piece. A construction using all pieces with k holes can easily be transformed into a construction made from all pieces with n - k holes. All rows in the table are symmetric.

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