A simple braid example

Advanced methods and approaches for solving Sudoku puzzles

A simple braid example

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):

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`+----------------------+-------------+-----------+| /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:

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`+----------------------+-------------+-----------+| /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-: Show
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`   *             |    |               |    |    *        *        |=()=|    /  _  \    |=()=|               *            *    |    |   |-=( )=-|   |    |      *     *                     \  ¯  /                   *    `

"If one is to understand the great mystery, one must study all its aspects, not just the dogmatic narrow view of the Jedi."

SpAce

Posts: 2016
Joined: 22 May 2017

Re: A simple braid example

Hi SpAce,

First of all, I do not want to vanish any thread.

I think that the braid analysis will be most widly usable for generating a puzzle from empty grid (scratch).

As I noticed in my thread that, when solving an empty grid, a braid analysis will definitely reduced the chance of backtracking or simply guessing.

So, this technique, if non-assumptive, will showing up its great power on generating a puzzle rather than solving a puzzle.

I am not sure but have strong feeling about my above statement.

Added: I was supposed that this technique is non-assumptive as per StrmCkr this post. But later on, noticed that braid analysis will also lead to zero state in this response as success rate is 89%.

R. Jamil
_________________________________________________________________
Simple maths:
2get and 2give create many problems.
So, just double it ...
4get and 4give solve many problems.
rjamil

Posts: 463
Joined: 15 October 2014
Location: Karachi, Pakistan