Locked Candidates or Box,Column, Row interactions


The following technique is only a tad more difficult than the first beginner tips. It involves applying the rules an additional time. This technique is often called Column, Box, Row interactions. I generally just call this tip Locked Candidates. Without regard to the choice of nomenclature, the technique is as follows:


If a candidate is already excluded from two of three boxes in any one row, then that candidate is locked into that row within the remaining box. Therefor, the candidate can be safely excluded from the other rows within that box.

If a candidate is already excluded from two of three boxes in any one column, then that candidate is locked into that column within the remaining box. Therefor, the candidate can be safely excluded from the other columns within that box.

If a candidate exists only in one row within a box, then it is locked into that box with respect to that row. Therefor, it can safely be excluded from the locations in that row outside of the box.

If a candidate exists only in one column within a box, then it is locked into that box with respect to that column. Therefor, it can safely be excluded from the locations in that column outside of the box.

After I introduce forbidding chains in this blog, this technique will be the only depth 1 technique. It is represented in a forbidding chain with just one strong link between either two cells, or two groups of 3 cells.


The Possibility Matrix

Before showing examples of this technique, it is proper to introduce the possibility matrix . Although many techniques can be utilyzed without considering the possibility matrix, the possibility matrix is a significant help for all techniques, including even Unique Possibilities. This is a simple concept. One merely enters all the possible candidates into each cell. When doing this, one applies the rule of exclusion - that a candidate can appear only once in a large container (box, column, row). Therefor, if a cell within a large container is already solved, that solved candidate is not entered as a possibility within any other cells in that large container. The following is an example of a puzzle with the possible candidates filled in.


possibility matrix

Please find below two examples of the locked candidates technique.


Candidates locked into a row, excluded from the rest of the box


locked sixes

In this example, the all the possible locations candidate 6 are highlighted in green. Note:

  • In row 9 the sixes are limited to two locations: g9,i9.
  • Both of these locations are located within box h8.
  • If a six were to occur within box h8 but not at g9 or i9, then row 9 would have no sixes possible within it.
Therefor, we can safely exclude sixes from all cells within box h8 but not in the ninth row. In this example, we can eliminate 6 from h7.

To those whom are observant, one can also eliminate that same 6 from h7 considering the 6's at d7,e7. In this case, the sixes in box e8 are limited to just two locations, both of which are in row 7. Therefor, if a six were to occur outside of box e8 in row 7, then box e8 would be left with no sixes possible within it.

In both of these cases, I would present this idea in a proof as follows:

  • Locked 6's at gi9 forbids h7=6. OR
  • Locked 6's at de7 forbids h7=6.

By the way, it is not unusual that the same candidate can be eliminated in more than one way.


Candidates locked into a column within a box, excluded from the rest of the box


locked sixes again

This is the same puzzle as the previous example, after eliminating the 6 from h7. Again we are looking at the sixes, highlighted in green. Note:

  • In column h the sixes are limited to two locations:h5 and h6.
  • Both h5 and h6 are within box h5.
  • If a six were to occur in box h5 outside of column h, then there would be no possible locations for 6 in column h.
Therefor, we can safely eliminate all the sixes from box h5 in columns g and i. In this case, we exclude six from g4,g5,i4,i5,and i6.

I would present this idea in a proof as follows:

  • Locked 6's at h56 forbids g45,i456=6.

After making these eliminations, note that row 4 will contain only 1 possible location for 6. Therefor, we can solve cell e4=6. This would be written in a proof as UP##, where ## would be the number of filled cells one can get using only Unique Possibilities from this point. In this case, it would be UP 29. The symbol % is used to mean the only place or candidate left. The next three steps then could be written as

  • e4=6%row
  • d7=6%row or d7=6%box
  • e7=5%box




11 Comments
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I don't understand how you can tell the number of unique possibilities (UP) is 29. Sorry if I seem completely dense here. I followed everything up to this point. Thanks
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The little boy asked his mother a question concerning the facts of life. She suggested that he ask his father. 'But mother, I don't want to know that much about it'. That is exactly how I'm feeling after this lesson. But I do respect the scholarship. Forge on.
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Wow!! Steve, thank you I often looks at your proofs when I am stuck on tough(very often), but have never understood why a certain number is 'forbidden' from a certain cell, (but just do what you tell me to!), and now I do!!! Your explanation is excellent!! I only hope I can follow the more complicated techniques as easily when they come along. Ta muchly!!
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Hi P from Ohio!
Perhaps the confusion is my fault:
In the example puzzle, there are already 26 cells solved (or filled in). After making the eliminations involving locked sixes shown in both examples, one can solve three more cells. This gives a total of 29 solved cells, and the shorthand More...
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Thank you. That makes sense. Steve, thank you also for taking time to explain how to read proofs. I look forward to eventually being able to understand the proofs posted in comments on the tough level!
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Steve, thank you so much for your thorough explanations. I agree with bluey. Before, I couldn't GET a clue about how to interpret your proofs or the logical solutions to the tougher puzzles, but now your 'Sudoku for Dummies' breakdown makes it so clear. Again, thank you. You make me proud to be a Buckeye.
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Steve, When some people write their proofs, they often use symbols like - - inbetween the box locations, and use one of these ~ Are you going to explain those at a later date? Thank you for all your hard work and easy explinations.
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Hi Angie (from Wisconsin)!
I intend to explain all of the symbols that I use. I should also be able to explain most, if not all, of the symbols emloyed by others.
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Steve, How did you develop your solving skills to such a level? Do you have a mentor...or do you devise these solutions in that clear thinking brain of yours? I'm so amazed at your know-how!!
18/May/07 12:45 AM
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An interesting result here is that when you eliminate 6 from H7 in the Candidates locked into a row, excluded from the rest of the box example, you now have locked 6's at H56 forbid G45=6, I456=6, leaving only E4=6 in Row 4. From there you can get d7=6.
03/Jul/07 4:57 AM
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Okay, here is a really dumb question -- since i am just now signing on to this site -- How are the boxes (regions? "gray areas" in the newspaper? numbered. I can get rows and columns but I'm confused about finding a specific box.
Help! Thanks,
18/Apr/09 5:44 AM
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