# Progressions of polyominoes - Advanced

1. Aug 19, 2010

### FaustoMorales

We define a progression of polyominoes with n terms as a sequence of n polyominoes, starting with the single square (the monomino), such that every shape is obtained by adding a square to the previous polyomino in the sequence.

Conjecture 1: For every odd integer n > 3, there is a rectangle-free partition of the nxn square containing the last n terms of some progression of polyominoes with (3n-1)/2 terms.

For example, the 5x5 square can be partitioned into a rectangle-free collection of shapes with sizes 3,4,5,6 and 7 corresponding to the last 5 terms of some progression of polyominoes with 7 terms.

Please post your solutions as you get them! The floor is now open for the first world record: the 5x5 case.

Conjecture 2: There exists an infinite Master Progression of polyominoes MP such that for every odd integer n > 3, there is a rectangle-free sequence of n consecutive terms of MP that can partition the nxn square.

Once you have solved the 5x5 case: Can you solve larger cases consistently with conjecture 2?

Last edited: Aug 19, 2010