Pentacube Reptiles

A pentacube is shown below where it forms a number of boxes. If we combine the 2x3x5 and the 3x3x5 we get a 5x3x5 and if we now join this to a 2x5x5 we get a 5x5x5 cube which shows that this pentacube is rep-5³.

There are certainly other values of n for which this is rep-n³ but no detailed analysis has been done.

The P and L pentominoes are rectifiable with the P forming rep-n² for all n and the L rep-n² for all n>3. The corresponding solid pentominoes are thus rep-n³ for the same values of n.

The Y pentomino is rectifiable for many values of n (see the reptiles page) the smallest of which is rep-9². The solid Y-pentomino is rep-n³ for all these values but is also rep-43, rep-53, rep-83 and rep-123. The bottom diagram below shows two ways of producing a depth 4 dekomino which can be fitted together as shown produce an eightfold copy of the pentomino. Since this is depth 4 two of these will form the required replication.

The diagram below shows how to extend a rep-n2 by 10 (or 15) provided a 10xn (or 15xn) rectangle exists. This extension can be used with the depth 4 constructions above to make replications for n = 28, 32, 48, 52

The leaves the only open cases for rep-n³ as n = 13, 14, 17, 18, 22, 23, 27, 33, 37, 38, 42, 43, 47, 53, 57, 58, 62, 67, 68.

The smallest replication for the U pentacube is rep-33

The smallest replication for the V pentacube is rep-63

rep-83