A simple braid example

Advanced methods and approaches for solving Sudoku puzzles

A simple braid example

Postby SpAce » Wed Nov 22, 2017 1:29 am

I recently started experimenting with braid analysis a little. It seemed like an interesting and even intuitive, if questionably useful, technique to try to integrate into my manual system. So far it's mostly paid off as an additional layer of validation and error-checking as well as for better visualization of the puzzle structure, but it has also led to a few simplified deductions in some puzzles. It hasn't been terribly useful, but just enough that I've decided to keep experimenting. Instead of using complicated tables as in the Sudopedia examples (seems tedious, error-prone and unintuitive), I just added simple strand-markers within my pencil-marks, which keeps the overhead pretty light.

Here's a simple example in which we're interested in the leftmost tower (I'm showing the strand markers for it only):

Code: Select all
+----------------------+-------------+-----------+
| /1    247      45    | 23  2357 37 | 8   9   6 |
| /8    267      56    | 4   257  9  | 57  1   3 |
|  35   37      /9     | 6   1    8  | 57  4   2 |
+----------------------+-------------+-----------+
| /7     36      356   | 28  28   4  | 1   56  9 |
| /2    /9      /8     | 5   6    1  | 3   7   4 |
|  45    46     /1     | 9   37   37 | 2   56  8 |
+----------------------+-------------+-----------+
|  6    /8      /2     | 7   4    5  | 9   3   1 |
|  34    5       34    | 1   9    2  | 6   8   7 |
| /9    /1      /7     | 38  38   6  | 4   2   5 |
+----------------------+-------------+-----------+


As shown above, I first strand-marked the solved cells only, because it's easiest and sometimes enough. From that we can quickly see that the tower is either a ZZZ-rope or a ZZN-braid type, but we don't know which. That doesn't help much so I marked the candidates next, which takes a bit more effort and carefulness:

Code: Select all
+----------------------+-------------+-----------+
| /1    /2/4/7  /4\5   | 23  2357 37 | 8   9   6 |
| /8    /2\6/7  \5/6   | 4   257  9  | 57  1   3 |
| \3/5  \3/7    /9     | 6   1    8  | 57  4   2 |
+----------------------+-------------+-----------+
| /7    \3/6    \3/5\6 | 28  28   4  | 1   56  9 |
| /2    /9      /8     | 5   6    1  | 3   7   4 |
| /4\5  /4/6    /1     | 9   37   37 | 2   56  8 |
+----------------------+-------------+-----------+
|  6    /8      /2     | 7   4    5  | 9   3   1 |
| \3/4   5      \3/4   | 1   9    2  | 6   8   7 |
| /9    /1      /7     | 38  38   6  | 4   2   5 |
+----------------------+-------------+-----------+


In this case that step does provide some additional information. We can easily see that the 3s follow an N-strand in all cases, which makes the ZZZ-rope impossible. That leaves the ZZN-braid as the only possibility, which tells us everything we need to solve the puzzle, as there are several box-columns with only one N-candidate (which must be true). For example, we now know immediately that r3c1=3, r4c2=3, r8c3=3, and r6c1=5.

I'd hate to rely on a braiding solution only, but in this case it's easy to check the result and, while doing it, to find a Y-Wing or an XY-Chain as a more traditional solution. Of course you can rather easily find them anyway (or simply guess), but in this particular case I found the braiding solution first and without really thinking or searching (as it was a more or less automatic result of the markup process). Please note that this is just a simple example of how basic braid analysis can be used as an optional supporting technique, nothing more. By no means is it necessary or even efficient here. (I haven't found more relevant examples yet.)

PS. Feel free to point out any mistakes in my analysis or markup. They're quite possible with my short experience with this technique.
SpAce
 
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Joined: 22 May 2017

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