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.

 

[LudusRex]

I find this conjecture to be quite intriguing. Polyominoes, which are geometric shapes formed by connecting squares at their edges, have been studied extensively in mathematics and computer science for their interesting properties and applications. The idea of a progression of polyominoes, where each shape is formed by adding a square to the previous one, is a fascinating concept that has been explored in various contexts.

However, when it comes to proving this conjecture, it is important to consider the complexity and potential limitations of polyominoes. While the statement may hold true for some cases, there may be certain arrangements or combinations of shapes that make it impossible to extend a progression of 6 terms to 8 terms while filling a 6x6 square. It is possible that there are certain "nasty-looking" cases that may require a different approach or additional conditions to be met.

In order to prove or disprove this conjecture, it would be helpful to first consider the properties and characteristics of polyominoes, such as their symmetry, connectivity, and possible combinations. Using computer simulations or visual aids, we can explore different arrangements and progressions of polyominoes to see if the conjecture holds true in all cases or if there are any exceptions.

Additionally, it may be beneficial to collaborate with experts in the field of polyominoes and combinatorics to gain insights and perspectives on this conjecture. By combining theoretical analysis with practical experiments, we can work towards a better understanding of the progressions of polyominoes and their potential to fill a 6x6 square.

In conclusion, while this conjecture presents an interesting and potentially valid statement, it also poses a challenging problem that requires careful examination and analysis. I encourage further research and collaboration in this area to shed light on the progressions of polyominoes and their fascinating properties.