Can All 6-Term Polyomino Progressions Extend to Fill a 6x6 Square?

[FaustoMorales]

Let us 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: Every progression of polyominoes with 6 terms can be extended to a progression of polyominoes with 8 terms so that the set of shapes thus obtained can fill a 6x6 square.

Please post any particularly nasty-looking cases and perhaps someone will be able to help. Good luck!

 

[loseyourname]

I'm not sure I'm understanding this. How would you extend a linear coupling of 6 squares into a 6 x 1 rectangle to fit into a 6 x 6 square? At least some of the obtainable shapes would have a linear dimension greater than 6, so they wouldn't fit inside that square.

 

[FaustoMorales]

some of the obtainable shapes would have a linear dimension greater than 6, so they wouldn't fit inside that square.

And we would not try to use those extensions containing polyominoes that don´t fit in a 6x6. The conjecture is that given any progression with 6 terms, we can find SOME extension thereof with 8 terms that can fit in a 6x6.

 

[loseyourname]

Okay, I get it. I thought you were saying all obtainable shapes from the 6-term forms would fit.