Sudoku: Fill & Download for Free

Sudoku puzzles are constraint systems. You need to find the missing numbers that satisfy the constraints given by the provided numbers, and the rules of Sudoku.A given Sudoku (or Sudoku-like) puzzle could fall into one of three categories:Over-constrained. Such a puzzle does not have a solution. A proper solver would be able to prove that a given puzzle has no solution.Properly constrained. Such puzzles have a single solution. A proper solver would be able to find this solution.Under-constrained. Such puzzles have more than one solution. A proper solver would be able to find at least one solution. If it doesn’t stop at the first solution it finds, it should be able to find all possible solutions.Proper Sudoku puzzles fall in the middle category. A reasonable Sudoku solver should be able to handle Sudoku-like puzzles that fall in any of the three categories.I wrote a very tiny Sudoku solver that runs pretty fast on most computers. For under-constrained puzzles, it only reports the first solution it finds. It’s a very straightforward recursive algorithm. The main recursive step is this:Find the first open square in “reading order.”If there are no open squares, report SUCCESS, print the solution, and exit.Find what digits are still valid options for that position.If there are no valid digits for that position, return FAILURE and backtrack one level.Otherwise, for each valid digit, put that digit in the open square and recurse. If all of them failed, return FAILURE and backtrack one level.If we backtrack to the beginning without seeing SUCCESS, the puzzle has no solution.That’s a fairly simple algorithm, and it’s surprisingly effective. You can implement some rather inexpensive bitmap tracking structures to make it cheap to ask “what digits are still valid options for this position,” but that’s an implementation detail.You can implement some optimizations, such as finding the “most constrained” open square rather than the first open square in reading order. That may improve the run-time for certain puzzles.You could try to implement the heuristics humans use to solve a Sudoku. Folks have done that. I’m unconvinced it’s worth the effort. Every time I’ve tried to speed up my own Sudoku solvers, I’ve found that only very simple tweaks to bitmap management help consistently. For some puzzles, I get a boost from one step of lookahead, and that’s about it.Ultimately, though, it’s not hard to write a Sudoku solver.All that said, that doesn’t mean everyone who can program can write a Sudoku solver. My wife had a very pretty looking Sudoku game. Nice artwork, decent puzzles. It also had a solver mode, where you could give it an unsolved Sudoku, and it would try to solve it.Except… it would report “no solution” for certain difficult Sudokus. I took those and fed them to one of my own Sudoku solvers, and it found solutions to each of them in milliseconds. I mean, like 10 - 15 milliseconds.I don’t know what the programmers implemented for that Sudoku game’s solver, but it wasn’t very good.

The first thing this mathematician notes is that Sudoku has nothing to do with numbers. That’s right the digits [math]1[/math] to [math]9[/math] are used as mere symbols. You might as well use A to I. Or a collection of nine Egyptian hieroglyphs. Or pictures of your nine closest relatives.Secondly Sudoku is a fairly simple example of a Constraint Satisfaction Problem (CSP). One that is easily solved by straightforward (computer) algorithms. Any particular Sudoku puzzle is not an interesting mathematical problem, although finding minimal constraints (least number of squares filled in) for a unique solution is a mathematical challenge.CSPs typically have easily discoverable sub-constraints that are necessary but not sufficient to solve the problem. These sub-constraints show up as Sudoku “rules” that people use to gradually fill in the squares. Easier puzzles have rules which enable simple inferences to complete the solution. Medium puzzles typically require (human) solvers to collect binary options (often written “in pencil” on the puzzle) until some deductive inference leads to a cascade of resolutions of these options. Harder problems may require a non-deductive “guess” to see what its implications are.Creating a Sudoku puzzle that has a unique solution is a harder problem. Assuming [math]P\neq NP[/math], there is no polynomial-time algorithm[1] for generating general puzzles with unique solutions. But the space of [math]9\times9[/math] Suduko puzzles is sufficiently small, and computer solvers are sufficiently fast, that you can generate puzzles by computer without too much difficulty. It is this fact together with the easily recognised sub-constraint “rules” that make Sudoku puzzles so widespread and popular.Most of solving a Sudoku puzzle is not particularly mathematical. The inference “rules” can be formalised as logical deductions but it doesn't add much for the solver. My conclusion is that Mathematicians do not solve a Sudoku puzzle in any manner that is substantially different to how anyone else solves that puzzle.Footnotes[1] P versus NP problem - Wikipedia

Donald Trump, definitely.Because I would have no qualms about immediately getting up and leaving.With Melania, I’d feel compelled to spend a few minutes in some respectful social conversation, and I’d be losing some time I’d rather spend solving sudoku puzzles.With Trump, I’m gone immediately.