Calculate the number of free n-polyominoes

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In summary, the conversation discusses the availability of a formula to calculate the number of free n-polyominoes. One person asks for confirmation of its existence and if not, the reason for its unsolvability. They also mention Martin Gardner's mention of the problem being unsolved in the past.
  • #1
Aeneas
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As far as I can make out, there is no formula available to calculate the number of free n-polyominoes, only bounds. Can you please confirm whether this is the case. If there is a formula, could you please point me to it. If there is not, is the problem just unsolved or has it been shown in some way to be insoluble?

At first sight, it looks as if it ought to be a simple formula to find, so if one cannot be found, is it possible to identify the reason why?

Thanks in anticipation.
 
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  • #2


I think Martin Gardner has written in one of his many books that the problem is unsolved. That was a while ago though, back in the 60ies/70ies and a lot has happened since then. Typically though, most problems of this sort are difficult to solve.
 

1. How do you calculate the number of free n-polyominoes?

The formula for calculating the number of free n-polyominoes is 2^(n-1). This means that for every additional square added to the polyomino, the number of possible combinations doubles.

2. What is a free n-polyomino?

A free n-polyomino is a geometric shape made up of n number of squares, where each square shares at least one common side with another square in the shape. The shape can be rotated and flipped without changing its overall structure.

3. How can the number of free n-polyominoes be useful in science?

The number of free n-polyominoes is important in fields such as computer science and material science. In computer science, these shapes are used in coding and game design. In material science, they can be used to understand the properties of certain materials and their behavior under stress.

4. Are there any real-world applications for free n-polyominoes?

Yes, there are many real-world applications for free n-polyominoes. They can be used to study crystal structures in chemistry, to model the growth of plants and animals in biology, and to design efficient packaging or tiling patterns in engineering and architecture.

5. Are there any limitations to calculating the number of free n-polyominoes?

Yes, there are limitations to calculating the number of free n-polyominoes. The formula only applies to free polyominoes, which means that each shape must be able to rotate and flip without changing its structure. It also does not take into account the physical limitations of creating these shapes in the real world, such as the size and number of squares that can be used.

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