We define a(adsbygoogle = window.adsbygoogle || []).push({}); progression of polyominoes with n termsas 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 everyoddinteger n > 3, there is arectangle-freepartition of thenxn squarecontaining thelast n termsof 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 firstworld record: the 5x5 case.

Conjecture 2:There exists an infiniteMaster Progressionof polyominoesMPsuch that for everyoddinteger n > 3, there is arectangle-freesequence ofn consecutiveterms of MP that can partition thenxn square.

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

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# Progressions of polyominoes - Advanced

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