Cookie Consent by

Replications with pentominoes

(a problem suggested by Dario Uri)

It is known that the 12 pentominoes cannot be used to form 4 identical shapes. (Problem 88 in Golomb's book.)

Dario Uri has proposed (on the newsgroup it.hobby.enigmi) the following similar problem: let's take two full sets of pentominoes and let's exchange two pieces between the sets.

Could now each of the two sets form 4 identical shapes?

Below is one of the possible solutions, obtained exchanging pentominoes N and Y. Livio Zucca and Dario Uri found almost all the solutions.

To solve this problem (in the trivial way) I generated all the 3,436,576 15-ominoes, I replicated them 4 times and I searched for tilings. In this way, I obtained 79 half-solutions.

By combining compatible half-solutions I found that 6 exchanges are possible and lead to a total of 10 solutions.