Some years ago I wrote a "brute force" Sudoku solver which goes row by row, finding all possible solutions. It's in the Library Archive. The Integer programming method runs much faster. I also tried using Reduce with Unequal constraints as well but ran out of patience waiting for it to return.

It would be nice if Reduce applied this technique automatically.

using Unequal constraints or Integer Programming?

FindMinimum takes about 0.1 sec. Reduce does not return a solution after 10 minutes.

yes, I tried Reduce[ allcons, vars, Integers ].

Very cool. I see this post is trending now on Reddit-Programming: https://redd.it/5h1xgq There is a comment there mentioning MS-OML. Just for the sake of comparison i give their code that solves the problem. I wonder if we can learn anything form it.

Model
[
    Parameters[Sets,I,J],
    Parameters[Integers,d[I,J]],
    Decisions[Integers[1,9],x[I,J]],
    Constraints
    [
        FilteredForeach[{i,I},{j,J},d[i,j]>0,x[i,j]==d[i,j]],
        Foreach
        [{i,I},Unequal[Foreach[{j,J},x[i,j]]]],
            Foreach
            [{j,J},Unequal[Foreach[{i,I},x[i,j]]]],
                Foreach
                [{ib,3},
                    Foreach
                    [{jb,3},
                        Unequal[Foreach[{i,ib*3,ib*3+3},{j,jb*3,jb*3+3},x[i,j]
                    ]
                ]
            ]
        ]
    ]
]

Don't shoot the messenger. I just reposted it here verbatim for easy access by Community folks. I do not understand MS-OML language at all. Perhaps some knowledgable folks would be able to comment.

Much clever thought went into your solution, thanks for sharing! I did not understand at first, what and why was happening in your code and i had to refresh my memory on numerical optimization. Understanding now the principle of your algorithm (maybe it cannot be called an algorithm because it is just setting up the maths to be solved by the solver in 1 step), i actually cannot think of any other alternative sensible way of solving the sudoku puzzle on a computer. I believe that there are many different ways (actual algorithms!) which solve the sudoku puzzle on a computer, e.g. working with permutations and such. But i always wanted to fully understand just one way of getting there, and if it is a mathematical way, the better. And yours is a purely mathematical way, solving a huge system of equations!!

To not lose my train of thought, i've edited your *.nb-file (minor code rearrangements and improvements, added explanations) and am sharing it here as well, please see attached.

Hope this helps, God bless!

Attachments:

sukoku.nb

Yep, that is a true algorithm (original by whose author?). Pretty concise. Solves even faster than the mathematical BILP solution. And it also spits out all other solutions, if there are more than 1 solutions to the puzzle. You can substitute the function NestWhileList to see the iterations and how it generates the alternate solutions for the output.

A one-liner! Absolutely mind-blowing. Uses multiple multi-nested pure functions and only basic vocab items. I once tried to dissect/modularize the code and create an annotated 13-liner from it, but failed to do so big time. If i do not understand why the code does the what, to what end, it is impossible to …erh… understand it, duh.

So never mind the dissection or modularization. I don't understand the other Sudoku algorithms in C, C++, Matlab, Python, JavaScript, etc (which are 13-liners or longer) either, and i am not interested at all in understanding Sudoku solving algorithms because I am ignorant hehe. I only understand Frank's method; too bad that it spits out only one solution to the puzzle when alternate solutions exist.