How do you spot regular fish? An effective radar is the set of markings that you saw in the blank lines of the “Complementary Swordfish” panel of the last post, repeated here:
Looking from left to right, I marked the selection of c1 as a possible member by a vertical bar in a blank row, r4. I also noted the cell positions claimed by c1 with “plus” characters in blank column c4. Then as I moved across and found columns with common positions, I marked c3 and c9 as selected, without having to adding any “plus” positions. This trial marking often fails. If it had failed in this case, I would have started over with c5.
For lack of a better name, let’s call this spotting method the blank line tally. It is actually a visual version of the suset enumeration of the 10/25/11 post, applied to locked sets of fish lines.
An alternative fishing technique is reported in the www.sudocue.net solving guide. It is based on the fact that original fish lines and the opposite complementary fish lines contain nothing but member candidates. To quote, “If you are able to draw a line for each missing digit either horizontally or vertically in such a way that no two lines cross any of the remaining candidates, you are allowed to remove all digits not covered by a line.” It’s a snap to draw these lines when the fish are known, but absent that, one can only guess which candidates to draw lines though first. Repeated trials are not very practical, because chances are small that a fish is present. You could say that our blank line tally is a way of finding the right sudocue lines.
Finally, those who work with numbers better than panels might like to try another suset enumeration technique I once used for finding regular fish. I used a form on which the positions along lines were represented by numbers. My fish form for the above panel is shown here. One form is filled out for each panel number.
The column swordfish is very evident in the three repeated numbers 268. The task is to find n lines containing only n numbers, the same as the blank line tally and suset enumeration . The form works very well. Looking for a fish among the rows, for example, we quickly see susets of 357/2678, 28/1239, and 59/2578, but easily see this is going no further.
Although my fish form revealed tougher fish very well, I gave it up in favor of the X-panel, and the blank line tally. Besides being adequate for regular fish, and working nicely for X-chains and loops, the X-panel also handles the weird fish we saw at the aquarium. The fish form was a failed me there.
Denis Berthier, in The Hidden Logic of Sudoku, details a technique common in Sudoku computer codes, of mapping the normal numbers on row vs column of the Sudoku grid, to a rows on number vs column grid, or a columns on number vs row grid. Such mappings allow the algorithms these codes use for finding locked subsets to be applied to regular fishing. For example, on a numbers vs rows grid, the numbers line (row x) would contain the second column of the fish form above as candidates in cells 1 through 9. The candidate positions of the other 8 numbers would be in the other 8 rows of this grid. Thus fishing on columns becomes the finding of locked sets on rows of the grid.
A separate mapping to a numbers vs. columns grid would place candidate positions along rows, the first column of the fish form, into rows of the grid. You can derive this row from the original grid, or from the number vs. rows grid.
As to fishing with the above form for the Berthier mapping vs. X-panel, my choice for sysudokies is the X-panel. The mapping gets you a locked set fish form for every number at once, but the X-panel is easier to extract and is easier to check and correct. You might verify this for yourself by doing a Berthier mapping by hand. After the extraction, the X-panel gives you a lot more. It is there for X-chains, weird fish, and something I will get to later, pattern overlay.
While on the subject of fish finding , I have to share a swordfish grid that could only be generated by the sysudoku basic solving . It is in the early stages of a tough line marking, from Su-Doku Maestro July-September 2009, puzzle 22, a Niveau 8-9. On the fourth line fill of row 7, I look at the filled rows for an easy 5-wing or 7-wing and happen to notice the 7-swordfish. I mark fish on the grid sheet with my showy red fish icons, to help me avoid placing candidates in c357 of the remaining rows.
As this is posted, I am through the X-panel analysis and still have the unveiling of this dragon ahead of me. You can see that it did not cough up a single clue in box marking, and was teasing me in SW. If you have been following along, you can pick it up from here and see what I’m up against. If you get there first, you can tell me what, but don’t tell me how. I’m having fun, and I do wish this kind of Sudoku experience for you.
Now back to your homework:
If you took me up on doing the LA Times 5/14/09 line marking, there are at least two advanced removals you may have caught. One is a remote pair that denies 5 and 7 candidates to an unmarked cell. The other is a 5-chain skyscraper which makes a devastating removal. Ordinarily this would be found on the X-panel, along with the fish. But the characteristic parallel slinks of a skyscraper makes it easy to spot in line marking, as you place line marks and look for matching marks on parallel lines.
Skyscrapers seem to be haunting me ever since I blogged that sysudokies need not look for them in preference to other X-chains. Unfortunately, if noticed in line marking, this one destroys a 5-panel I wanted LA Times 10/14/09 to present to you. So next post, I will include for your comparison a line marking trace that assumes the skyscraper was missed and doesn’t show up until you generate its 5-panel. Play along with this one. You’ll like the way you look.