The discussion centers on the linear algebraic aspects of Sudoku puzzles, particularly the orthogonality of Sudoku matrices. Participants reference a paper that discusses whether a Sudoku matrix can be orthogonal, concluding that the squared Frobenius norm of any Sudoku matrix is too large for orthogonality. They also explore the implications of linear programming in solving Sudoku puzzles, suggesting that sufficiently constrained puzzles can be solved using real numbers instead of integers. Additionally, concerns are raised about Theorem 12 in the context of super Sudoku and its diagonal constraints.
PREREQUISITESMathematicians, computer scientists, and enthusiasts interested in the intersection of linear algebra and Sudoku, as well as those exploring optimization techniques in combinatorial puzzles.
I think he may be working within ##l(\mathbb Z) ## or ## l(\mathbb Z^{*} )## (working mod45 , IIRC, since ##45= \frac {n(n+1)}{2} ## ), though, I am not even sure Sudoku matrices are a subspace; origin not likely included. I was kind of lazy to just drop this here without checking the details before :(. Altho, given the fact that all entries are positive, there are no orthogonal elements in whichever subspace he chooses, by most defs. of inner product I can think of.StoneTemplePython said:I didn't understand the field being used here.
In particular part 9 "Orthgonality Discussion", which says “ The answer to this question is not known yet" -- i.e. whether a sudoku matrix can be orthgonal.
If we have an inner product, then it is very easy to show that the squared frobenius norm of any Sudoku matrix is far to large for it to be orthgonal. (Why? its minimum entry value is 1 and there is a 1 in every column -- all columns also have some number bigger than 1...).
If we don't have an inner product or anything "close" then I really cringe when people use a term like "orthogonal".
Just curious: I have found I can solve ( "paper and pencil", no strategy or explicit method) certain Sudokus well, but not so others. Do you think a classifier method may help me understand some traits in each group; the solvable and the non-solvable?StoneTemplePython said:Yea, as it happens my preferred way to solve sudokus is to use linear programming.
An observation I've had but kind of forgotten about, is everything that is sufficiently constrained to have one answer (i.e. the puzzles that make it to the general public), can be solved with the relaxation of using reals in your LP, as opposed to the integer domain it seems to require. I was poking around for something about totally unimodular submatrices or whatever.
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btw Theorem 12 has an issue --- in some forms of super sudoku there are constraints along the diagonal (and anti-diagonal) which would necessarily make the trace constant.
No, I mean solvable by me. And, while classifying may take away some of the fun/mistery away, I am interested more in learning ML methodology. I guess I would enter them as a 2D array.StoneTemplePython said:To be clear, do you mean "solvable" by you, or actually valid problems with feasible solutions? If the former, I guess it could be kinda fun to pass it to a classifier. Typically the way I think about problems and "machines think" about problems ends up being a bit different though. (I think it would be kinda fun to overfit some massive deep net on a few sample puzzles and publish a bunch of nonsense about it afterward though.)
I guess it comes to how much Sudoku you do, and whether computational insights will take too much of the mystery out of it.