How to Solve Sudoku Mathematically

I took up solving sudoku puzzles a year ago out of sheer curiosity, as to why are people hooked to it. Guess what, I got hooked too. Of all the riddles and puzzles I have come across, I find sudoku puzzles unique in the challenge they offer. This puzzle stands out due to its inherent simplicity and beauty.

About a Pure Mathematics Approach to Sudoku

After reading the title, you might think that I am going to expound some esoteric mathematical theory about how to solve sudoku using theorems. You can rest assured that I am not going to do anything of that sort! Although as a mathematical object, a sudoku grid is quite well-known as a Latin square. The reason I don't intend to go into esoteric mathematical theory for solving sudoku is, that it would take all the fun out of solving it for the non-mathematical background enthusiasts. Also, I will have to talk about combinatorics, permutation group theory, equivalence classes and algorithms which will make this article a treatise in mathematics. For those of you who have a graduate level understanding of mathematics, it will make sense, but to the people who are not in to mathematics, it will only scare them away. So, for those of you who are into hardcore mathematics, I will just share the names of two prime mathematical approaches. One approach is called 'Grid Analysis' while other one is puzzle solving using initial given values. Also, for mathematical wizards, let me say that sudoku puzzles have been certified as a NP-Complete problem. There are many more mathematical approaches assisted by computer programming.

Tips to Solve Sudoku Mathematically

My approach to solving sudoku is what one would call experimental. I try out things and follow one rule: 'Do not make the same mistake twice'. This approach can drive you towards your own improvised technique of solving sudoku or any other puzzle. I assume that you know the basic rules of solving a sudoku. If you don't know them, then just remember that it's all about filling a 9 x 9 partially filled grid with numbers from 1 to 9 once each; such that every row, every column and every 3 x 3 box of numbers has all the numbers filled once. Here I share whatever little I have learned after solving my own share of sudoku puzzles. It may give you some hints on how to solve sudoku puzzles quickly and let you develop your own technique of solving sudoku. I sum it all up in just two main points.

Single Out the Loners & Strike Them Out
Every square in the grid has nine possible choices of numbers, from which eight need to be eliminated. First part is finding the loners which are most obvious choices left to fill in a row, column or box. They are the squares which have rest of the eight choices already eliminated due to filled numbers. Tracking these loners is thus the prime job. This single rule is enough to get a solution.

Eliminate the Impossible Choices
Solving sudoku is all about eliminating choices and possible permutations, until you are left with what is inevitable. One technique is to maintain a list of possible choices for every square on a separate piece of paper and striking out the numbers as each square is filled. For every square, check out the numbers that have been already filled in the box to which square it belongs and also the row or column to which it belongs. If you find two squares with the same possible choices in a box, then you can experiment and check out what pattern forms through filling numbers in one. If it works and makes sense, then keep it and if it doesn't, eliminate it. Fill the number in other square in that case. These are the kind of tricks that you will have to come up with to solve an advanced sudoku cube.

People may disagree with me when I say that too much of a pure mathematics approach would take the fun out of sudoku. All that I meant was, that it's not the most popular approach due to lack of expertise in mathematics. Solving every puzzle is about how one deals with an unknown. Approaches can differ but as long as the end result is a solved sudoku puzzle, nothing else matters.