7 Apr 2008 @ 10:59, by John Grieve
Dialectical analysis of the Sudoku puzzle
A sudoku is a puzzle composed of 81 individual squares or cells, which join together into 9 larger squares of nine cells each. It can be looked at also as an array of nine columns and nine rows, each composed of nine cells.
The rules are what defines a game or puzzle and they create the contradictions in it. The basic rule of sudoku is that all the numbers 1 to 9, in any order, have to be in every row, column and square. But the defining rule or characteristic is that no number can recur in any row, column or square. Thus the main opposites here are “Identity” and “Difference”
This puzzle is constructed as a logical deduction puzzle whereby it should be possible to deduce all the absent numbers from the ones that are given, one by one, serially and piecemeal.
Now, just as the logical deductive process focuses on the rule that the numbers must be different, and uses that fact to deduce the outstanding numbers in the grid, so contrariwise, the dialectical method uses the opposite fact that in some parts of the grid, not subject to the rules, the numbers will be the same, will recur. This will give rise to certain characteristic patterns which can be used to intuit, guess and deduce the complete picture, holistically.
There are many patterns that arise in this way, but I will only mention three to illustrate my point.
Firstly, numbers cannot recur on rows, columns or in larger squares because of the rules. But they can and do recur when counted along diagonal lines. Thus on a diagonal across the whole grid it is sometimes possible to count three or four recurrences of the same number for example 4 or 7. Every diagonal is of a different length and there are sub-patterns within them that will reward study.
Secondly, what I call knight’s-move pattern. In many cases this pattern, created directly by the rules as a natural consequence of them, follows exactly the move of a knight in chess, and the same number recurs in the next column or row in this position. However, the rule that no number can recur in a larger square sometimes affects this pattern, and the same number recurs in a slight variation of the true knight’s-move, in an extended version.
Thirdly, it is noticeable that when a number is in a small square within a larger square, as we move from one column to the next, or one row to the next, the position of the same number is usually different, relatively, but not always. Thus, for example, in the top large square a 3 may be in the top left-most small square. In the next large square underneath it may be in the middle small square, while in the bottom large square it may be that it is in the right-most small square.
These and other patterns can be observed and learnt. When doing a sudoku you can use knowledge of these patterns to intuit, guess and deduce the overall distribution of the numbers throughout the whole puzzle, rather than just using a mechanical, solely deductive process to solve the puzzle piecemeal. These are examples of the Taoist and Confucianist methods of solving problems that I have written about elsewhere. Of course, it is possible to combine the two methods to get the best result.
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