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I have an open question about invertible matrices in the Sum Thing puzzle which I will ask in a separate topic in the forum. I first wanted to define the math of Sum-Thing as I see it. I'll be glad for any comments or additions. I will try in the future to attach some pictures to make this more understandable to those that do not swim naturally in linear algebra.

Each Sum thing puzzle can be turned into an (M x N+1) matrix which represents a set of M equations corresponding to M distinct sum thing areas defined by N circles. (The N+1 is because there is an additional column of the actual M areas. Of course these set of equations know nothing about the constraints of the game. That is the range of possible numbers for the circles and the fact that the numbers must be integer and distinct. Still I find that it could be helpfull as it allows to identify separate regions and the connection between them, as well as the linear relation between different circle variables. (For instance when one tries to simplify such (MxN) matrices he could get for instance that the value of circle 1 must be the value of circle 2 + 5... ).

I think the mathematical approach is especially helpful for puzzle makers as an extra tool to design interesting layouts before thinking at all about the actual values of the sum thing areas to be defined.

It is interesting to take the reverse route and start with a matrix and find the a relevant layout. Note that not every matrix can be converted to a 2D layout of a puzzle with no overlaps between the different something areas or crossing borders (not at circles). So that is already fun trying to understand what in a Matrix makes it possible to transform it to a layout of a something puzzle.

I found of special interest the sub-class of NxN matrices that can be inverted. This correspond to puzzles with N circles and N something areas, such puzzles have a unique solution no matter what are the N values you choose for the something areas (Of course a particular choice of areas does not guarantee that the unique solution of circle values will all be in the puzzle range, or integers , and without repeats...). I thinik puzzle layouts corresponding to invertible matrices are interesting becuase they will have a unique solution no matter how you define the range, and thus you can find a possible set of areas for the puzzle simply by filling distinct numbers from the range in all the circles. I also think such layouts are interesting as they can be used as sub-sections of larger puzzles that can be solved independently and serve as a starting point for the larger puzzle, or interact several such areas...

I'm heavilly addicted to sum-thing puzzles