Rectangles can be made for all even values of n. For odd n the situation is quite comples. No infinite strip of width 5 can be made with N pentominoes alone and so there will be a maximum value of n for which the only rectangle of the correct area is 5 x prime. Wider strips with only Ns are possible and in fact a strips of width 11 and 15 can be made (these were found by Torsten Sillke). This allows us to obtain the diagrams below. Since the width 10 and width 15 rectangles show that all values of n which are multiples of two or three are possible, rectangles with such values of n are ignored in the diagrams below.

These width 10 constructions give n = 2k

These width 15 constructions give n = 3k

These width 11 constructions give 11k - 1 for k > 2

These width 13 constructions give 13k + 1

These width 17 constructions give n = 17k + 5

These width 19 constructions give 19k + 7

These width 23 constructions give 23k + 14 for k > 5

These width 25 constructions give n = 10k + 3 for k > 3

These width 31 constructions give n = 31k + 19 for k > 1

These width 35 constructions give n =7k + 2 for k > 8


The rectangles for n = 25, 29, 31and 79 are by Patrick Hamlyn and the 57 and 63 are by Roel Huisman (these rectangles could also be made from the constructions above). Patrick has also shown that no solution exists for n=35 and that no higher values of n of the form p-12, where p is prime, exist.

The above covers a great number of cases and it is likely that all rectangles of width greater than 7 are possible. In fact the all values of n up to 1000 are covered by the constructions here.

Further generalisations are possible be modifying the above to provide downward extensions also.

Other width rectangles can be extended in a similar manner.

Width 19 and 23 -

Width 29

Similar constructions for rectangle widths 30j + 25, 29, 31, 35, 37, 41 and 43 would complete the analysis and show that rectangles are possible for all n other than those of the form prime-12.

The above only attempts to find all values of n for which rectangles are possible. Finding all possible rectangles would require somewhat deeper analysis of the problem. For example, the width 35 constructions give solutions for n= 7k+2 but some of these values of n would fit into a width seven rectangle and we should need to find which.