possible new technique

JB · Posted: Fri Jun 01, 2007 1:43 am Post subject: possible new technique

I doubt very much that this technique is really new, but I don't have time to read the descriptions of all the techniques, so I'm just going to share this briefly. I call it the rule of "PAIRS AND SINGLES," or "TWO OF THREE."

The technique is based on the observation that it's possible to create a Sudoku where every row OR every column within a each block contains the same triplet of three numbers, just in a different order, forming a kind of "perfect square." It's of course impossible to have both vertical and horizontal symmetry at the same time. But the same set of three triplets can be fairly very easily repeated in all nine blocks either vertically or horizontally. In fact this is the easiest kind of Sudoku to create. E. g.:

936 147 582
174 258 369
528 396 417
285 714 693
369 825 174
741 639 258
693 471 825
417 582 936
852 963 714

Note that in each column or row of blocks, the triplet is always in a different row or column. Note also that because there are only 6 total permutations of 3 numbers, and nine triplets are needed to fill the nine blocks, three of the permutations of the triplets must be REPEATED, so they occur TWICE, whereas the other three occur only ONCE. E.g., above, the permutations 396, 639, and 963 (black) each appear only once. The permutations 936 (red) 693 (blue) and 369 (green) each appear TWICE.

This is probably the mathematical basis of the rule.

However, a "perfect square" like this, though easy to create, is very rare.
The rule that's universally applicable comes in a different but related form. Notice that in the example above, if you look vertically rather than horizontally, TWO OF THE THREE numbers of the VERTICAL "triplets" are always the same. It's harder to see at first, but it's there. This rule, moreover, applies to EVERY Sudoku, both vertically and horizontally:

In EVERY Sudoku, within a vertically aligned column of blocks, TWO OF THE THREE numbers in each column within a given block are ALWAYS THE SAME. This is the PAIR. It's always together. The third number is the SINGLE. It's always in a different place for each PAIR. Rarely, it's in the same place as the pair, which makes a TRIPLET, as in the first example.

This also true for rows, i.e., looking at three blocks horizontally. Of course the numbers of the PAIRS and the SINGLES will be different, but the pattern is the same.
E.g. [please widen your browser window so the block is intact]

461 375 298 . . . . . In this example, horizontally, the pairs are 61, 75, and 28.
238 691 574 . . . . . The singles are 4, 3, and 9.
597 248 613
156 439 782 . . . . . In the first three COLUMNS, there are triplets: 425, 369, and 187.
723 856 941 . . . . . In the middle COLUMNS, the pairs are 32, 79, and 18, singles 6, 4, and 5.
849 127 365
375 984 126 . . . . In the last three ROWS, the pairs are 37, 49, and 26.
682 713 459 . . . . The singles are 5, 8, and 1.
914 562 837

Knowing which numbers are the pair and which is the single is very useful in the difficult middle phase of solving.

JB

Ruud · Site Admin

Welcome, JB

You can continue here: http://www.sudopedia.org/wiki/Braid_Analysis

Ruud
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Mosc_007 · Posted: Fri Jun 08, 2007 4:49 pm Post subject:

It seems the possable solution techniques for Sudoku never ends.

Charles