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I would like to raise the question what is the minimal number of circles in a sum thing puzzle layout that correspond to an invertible matrix (See part I topic for details)?

I currently found layouts for 7 and 8 circle layouts which should be published online in the future, I'm 99.99% sure that they don't exist in 5 circle puzzles, and could not find any for 6 circle puzzles even though I searched desperately for the past two days.

It would be nice to see more layouts of N circle N area sum thing puzzles that correspond to invertible matrices.

Matbe the answer to the question would be different if the different sum thing areas do not have to be in one plane. 3D sumthing puzzles... Will be glad to hear other thoughts.

I've finally found a 6x6 invertible matrix with a sum-thing layout. I'll leave it as a riddle in the forum for now.

Interestingly, if we allow to also specify the sum of the outer face, then it is possible with just 4 circles. The complete graph K4 is planar so it can be drawn, and it has 4 triangular faces, resulting in the matrix

Code:

which is invertible. A puzzle based on this might look like this:

Nice! I immediately look at your example as a 3D pyramid where you specify the bottom face as 9. I realize with more complex patterns it may not always be possible to look at a 3D representation. We should think whether to allow outer sums in sumthings

Cool that you brought it up.