Possibility Matrix - An Alternative Method To Solving Every Sudoku

As a Sudoku lover, you may have often wondered whether there is a generic way to solve Sudoku. Here I have tried to address the question of a generic way of solving Sudoku, by discussing solving methods and their shortcomings.

There are multiple approaches to solving Sudoku. Broadly, they can be classified into three:

(1) Manual methods (2) Computer-based methods, and (3) Hybrid methods.

The essence of solving a Sudoku lies in the thrill; so, computer-based methods are likely to cause the puzzle to lose its luster. Since computer-based methods are a 'kill-joy', at best, one could use them to verify the solution. And we'll see about Hybrid methods, later. Manual methods could be one of the following two approaches to solving a Sudoku puzzle:

(a) The Conventional (or Intuitive) method, and (b)). The Possibility Matrix method (A Mathematical/ Scientific Approach)

While the conventional method attempts to find a Cell for a given value (1 to 9), the Possibility Matrix method finds the right value for every one of the unfilled Cells. Despite its popularity, the conventional has the following shortcomings:

(i) If a puzzle has no answers, you would not be sure. Just because you are unable to get a solution, you cannot conclude that the puzzle has no answers; it may be that you have missed out on a possible answer. (ii) If it has an answer(s), you are not 100% certain of finding it (them). (iii) If the puzzle has multiple solutions, you may not know that there are multiple solutions, and may not be able to find every answer easily, if and when necessary.

(iv) If the answer you get is different from the one given by the source, from where you got the puzzle, you may not know if this is because the puzzle has multiple answers.

(v) If you are caught in a puzzle with a complex multiplicity of possibilities, you may not be able to solve it even if there's a way to solve it, because of the sheer complexity. (vi)You may never be able to determine a reasonable time-frame within which you could solve, as with all 'trial and error' methods of solving complex puzzles.

(vii) There may be no guarantee that you will, indeed, be able to solve a given puzzle by this method.

(viii) Since there are a lot of rules to be learned in solving,(some of which are complex), there is a high probability of going wrong, along the way.

(ix) And some of the strategies are very cerebral, and, hence, not for everyone. Many senior citizens and people with difficulty in learning multiple techniques and using the appropriate ones at the right times have often found it difficult.

Yet, the conventional method is the one followed by most Sudoku solvers, as many people enjoy the 'random walk' method, the way Crossword Puzzles are solved. But there is no reason why you should solve Sudoku the same way you would solve a language puzzle. One can learn this method, but perhaps after learning the Possibility Matrix method, for all the above reasons.

Incidentally, it is usually the Possibility Matrix method that is used by software that solve Sudoku. Though this may take a little longer to solve simpler puzzles initially, you will learn to solve faster, with experience.

In simple terms, the secret of Possibility Matrix is filling up each of the empty cells with all the possible values that can occupy it. This is called the construction of the Possibility Matrix.

This is the first, but most important, step in solving Sudoku.

Now, have a look at an example that will teach you to construct a Possibility Matrix, for a 2 X 2 Sudoku, so you understand the PM method easily.

Click below to access the example:

http://www.howtosolveallsudokupuzzles.com/olacc/o2pm.php

Now, try to construct the Possibility Matrix for the SAP puzzle given below:

http://www.howtosolveallsudokupuzzles.com/olacc/order2.php

(By the way, you will notice that this is the same as the puzzle above, except for a value in an additional Cell.)

You will also notice that we have already solved this by an intuitive method, but not all puzzles can be solved just intuitively. This is why we are now learning a method that will help us solve all Sudoku SAP puzzles. So, let the fact that we have already solved it intuitively not bother you.)

Isn't it fun getting the Possibility Matrix?

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The above is an excerpt from the free newsletter on “How To Solve All Sudoku Puzzles” published by Geostar Publishing & Services LLC.

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