Many people at this forum and in general are concerned with the way candidates are represented. The method offered here, bypasses the need of “pencilling in”. It is meant for solving on paper and works well up to the “hard” level and maybe beyond. I believe it’s a great help for seeing things differently. Everyone doing sudoku has his/her own idiosyncratic way which suits him/her best. However I think the approach presented here is more general than that and can be applied widely. The topic was published here before, but was graphically inferior and did not convey the notion properly. So I thought it might be a good idea to improve on it and post it once again.
The title of this topic is somewhat misleading. Not only does this method dispense with pencil marks. It also enables crosshatching and actual placement of candidates. This additional all-important capability entirely escaped my attention. Although I was doing it all the time, I was biased and it occured to me only recently.
So, to be precise, the title actually should read: “Another Way to Represent, Crosshatch and Find Placement of Candidates”.
While solving a puzzle, how does one know at a given time that he has exhausted all possiblities with “slice and dice”and he’s really stuck at a lower level? The answer to this is – he does’nt know. At least he’s not sure.
If done properly, (which is easy), the method described below, tells you exactly when to use more advanced techniques, if at all. It’s a matter of efficiency.
When I do sudoku, I do the usual slicing and dicing until I ‘m stuck or the going gets harder and slower. Then I take a plain sheet of paper and put down the candidate situation of the nine boxes as it is at this stage. The boxes are numbered from 1 to 9 in a left to right, up to bottom fashion and in the same way do I relate to each blank cell within each of the 9 boxes. Taking the following box of a puzzle as an example:
I notice that there are 5 missing candidates in this box. They are: 2 3 5 8 and 9, which means there are also 5 blank cells to be filled by the above candidates. So I relate or enumerate mentally only the blank cells from 1 to 5 in the same left to right, up to bottom fashion. Then, on the sheet of paper I put down the 5 candidates in 5 rows, each row for one of the 5 blank cells, in this way:
and start checking each cell, crossing out the candidates that are’nt possible for that cell. Then I scan the table horizontally to find naked singles and vertically for hidden singles. For example, if after examination, the above table of candidates looks
like this:
it means that nothing was achieved yet . So I go on to another box (not necessarily to the next one), make a table of its candidates and cross out those which are’nt feasible for its blank cells. Then I scan horizontally and vertically as before. Now, if I’m more lucky and the topography of the box looks like this:
it means that we have found 6 to be a naked single in cell No.1 and cell No.3 to be the only one which may receive a 4, that is a hidden single. So I encircle the 6 in cell No.1, and cross out vertically all other appearances of 6 in the remaining cells. Next, I encircle the 4 in cell No.3 and cross out horizontally all other irrelevant candidates for that cell.
Vertical elimination for naked singles and horizontal for hidden singles is not some kind of mannerism. It’s so because the naked 6 was found after scanning the table horizontally, so that value must be crossed out from all other cells vertically. On the other hand, the hidden 4 was found after scanning the table vertically and since it occupies a particular cell, all other values from that cell must be eliminated horizontally. Actually, this is how the naked singles differ from hidden singles. Moreover, by using this candidate representation, the hidden singles are no more hidden!
So our table now looks like this:
If possible, I then update the candidate tables of the neighboring boxes and see if a breakthrough can be achieved there too. The updating is very important. Not doing it accurately or not doing it at all, may result in lost opportunities, to say the least. Only in the final stages, when your puzzle falls apart like a house of cards, can you afford not to update. Of course we insert the 6 and 4 in their relevant cells of the puzzle which was our purpose in the first place.
By the way, instead of making candidate tables for boxes, you can do the same for rows and columns. But I found the box table ergonomically more suitable and solver- friendly.
Of course, since candidate representation is performed externally, the sheet of candidate tables must be used with puzzle at hand in combination.
Of special interest to the solver are pairs, triples and maybe quads. Fortunately, these too are quite visible in such a candidate representation. Even their orientation can be deduced quite easily. Finally, since candidate representation in this way is done externally, it saves us the clutter of “pencilling in” and leaves the puzzle clean and workable.
Now, let us have a closer look at the system, Given the following puzzle:
The slicing and dicing procedure takes us this far (or maybe a bit further, but for our purpose, it’s immaterial):
And the candidate tables of the 9 boxes are:
Which after scrutiny becomes:
Whereupon:
r2c4 = 3 (naked single).
r4c5 = 3 (naked single)
r8c8 = 7 (naked single).
And the tables are updated accordingly:
After having inserted our findings in the puzzle, we must update the tables to reflect the recent changes:
Whereupon two hidden singles and three naked ones become apparent:
r1c8 = 6 (naked single).
r3c4 = 7 (hidden single).
r7c5 = 7 (hidden single).
r7c8 = 1 (naked single).
r8c4 = 5 (naked single).
Please note: As we update the tables, new singles may pop up. They are easy to spot. Of course, at this stage we could have done some slicing and dicing as well or prefer some other technique. (Actually this puzzle embodies an x-wing pattern too). So the general idea is:
1. While going is possible – do some slicing and dicing.
2. Got stuck? – resort to candidate topography tables and update changes.
3. While going is possible – do some more slicing and dicing and update tables.
4. Got stuck again? – try x-wing, swordfish or whatever other technique at your disposal, untill puzzle is completed.
Thanks for your attention, and do have a try at it
. Your comments,
and
will be much appreciated.Kibitzer
This new version doesn’t seem quite as action-packed as the first, but I must say the graphics are pretty spectacular.
