Pairs Triples Quads

  • As this page one visits,
  • A grace I elicit.
  • Please scroll to the comments
  • And read 'most a sonnet.
  • Find there a few staged words,
  • To speak of, I implore.

For all of the rest of the techniques presented, it is assumed that one is using possibilities. The following idea can be expressed in many ways. One way to understand it follows:

Naked Pairs

Suppose that two cells within at least one large container (box, column, row) have exactly the same two candidates left as the only possibilities for these two cells. Then, one can exclude those two candidates from all the rest of the cells within that large container(s).


  • A cell outside of those two cells
  • That cell is also within the paired cells' common large container(s)
  • Let that cell solve as one of two paired candidates
  • Then both of the original cells would be forced to equal the other candidate.
Since no candidate can appear twice within the same large container, this is forbidden.

One may notice that most proofs of ideas are proofs by contradication. This will be the case with most techniques. To eliminate a candidate in the possibility matrix, one generally must show that cell X = candidate N violates the rules.

The following is an example of Naked Pairs .

Naked Pair 59

In this example,note that cells h3,i3 both have only 5,9 left as possibilities. Since cells h3,i3 are both contained within row 3 and box h2, we can exclude 5's from d3,f3,h1,h2,i1. We can also exclude 9's from h1,i1. In a proof, this could be presented as follows:

  • Pair 59 at hi3 forbids df3,h2,hi1=5 and hi1=9.
Although the above presentation of proof lingo is my preferred style, one might also see other ways to present the same idea such as:
  • hi3=Naked Pair 59=>df3,h12,i1~5;hi1~9.
  • NP 59 = hi3 => df3,h2,i12<>5;hi1<>9
  • hi3={59} forbids df3,h2,i12=5 and forbids hi1=9.
The choice of proof presentation is not standard, so one must try to logically decipher each proof writer's codes.

The same idea can be extended to any number of cells contained within a common container. Thus, if for example we had:(the grid above does not illustrate the following example.)

  • a1 = 123
  • a2 = 123
  • a3 = 123
One could safely conclude:
  • Forbids b123,c123,a456789=123
One idea that bears mentioning here is that it is not necessary that all the possibilities be fully represented, for example:
  • a1 = 12
  • a2 = 23
  • a3 = 13
  • Forbids b123,c123,a456789=123
Hopefully the reasoning for this is clear. In either case, one could present this idea in a proof as:
  • Triplet 123 at a123 forbids bc123,a456789=123.
. The proof of extending this idea beyond two cells can be done many ways. The easiest way, in my opinion is to just think of it inductively: No matter what a1 equals, in the first example, there will be a naked pair in a23.

Hidden pairs, triplets, etc.

The next idea is really just a logical extension of the former. Consider any large container. If it is subdivided into two or more parts by naked subsets, then the parts of that container disjoint from those naked subsets must be inhabited by hidden subsets.

Alternatively, one can consider the following logic:

  • If a set of N candidates is limited to N cells within a large container.
  • Then those cells can contain only those candidates in the set
  • Thus one can safely exclude from those cells all candidates that are not members of the set.

The following is an example of a Hidden Pair .

Hidden Pair 46

In this example, note that the 4's in box b5 are limited to only two locations: a6,b5. Note further that the 6's in box b5 are limited to the same two locations. We can therefor forbid everything that is not 4,6 from cells b5,a6. The observant amongst you may note that we also have a Naked Quintuple with candidates {12389} at cells abc3,c56.

This idea could be presented in a proof as follows:

  • Hidden Pair 46 at a6,b5 forbids a6=123 and forbids b5=1389.

Again, the manner of presenting this step in a proof is also not uniform across provers, and sometimes not even uniform within a prover...

As with Naked Subsets, one can extend the idea to triplets, quads, etc. with Hidden subsets.

A caveat is required with both hidden and naked subsets, also called hidden and naked n-tuples. With all n-tuples, it is not required that all candidates appear in all the cells, as in the latter example listed above with candidates 1,2,3.

Hidden pairs are a favorite of mine to employ during set up. They are often easier to find before entering any possibilities.

Combining techniques

The next example of a hidden pair illustrates how one can combine techniques into one step. Consider the following partial puzzle grid:

Hidden Pair 39

Note that the 3's and the 9's in box e8 are limited to the two cells d9,f9. Thus we have hidden pair 39 at df9. Note however that b9=3 and hi9=9 are also still possibilities. Because of locked 3's at df9, and locked 9's at df9, we could also forbid b9=3, hi9=9. One could write:

  • Locked 3's at df9 forbids b9=3
  • Locked 9's at df9 forbids hi9=9
  • Hidden pair 39 at df9 forbids d9=2,f9=12
But, it is more efficient to merely write (and think):
  • Hidden pair 39 at df9 forbids b9=3, hi9=9, d9=2, f9=12

A complete rendering as to how this latter approach is more logical, in my opnion, will be presented later in this blog.

Indicate which comments you would like to be able to see

This past week has been extra-ordinarily busy for me. The weather has been comletely un-December-like, and the time I have had to devote to Sudoku has been, well, just about non-existent. Nevertheless, as I was driving to work one day early this week, a partial parody of the much parodied More...
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Twas the Sudoku 'fore Christmas, when throughout the grid,
Not a cell was a'solving, nothing to forbid.
The puzzle was printed and marked with care,
In hopes that such efforts a cell would prove bare.

The candidates were scribed each where they might,
But visions of an More...
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Fun, indeed! Steve, this poem leads me to believe you do have more time on your hands than your first post claims. If you can sit down and write such a poem with ease, I wonder then why you're wasting your skills on a Sudoku blog instead of investing your time on something more worth your while. More...
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Steve, great poem! My question regarding proofs that have been given on various puzzles is, what does 'start at 22 filled' mean? If the grid is labeled 1 - 9; a - i, then where is 22? You use this to start off your proofs as do others, ie: 12/18 on tough.
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Again thank you Steve, not just for your fine explanations but also for the wonderfully witty version of 'Night before Xmas'. This technique is something that I have actually been doing for a while now, without know it's name or even recognising it in your proofs!!!
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Hi Stella!
I usually write:

1) Start at ## filled - the given puzzle. Unique Possibilities to # filled. (UP #).

Step 1) The given puzzle has ## cells filled in at the start - or given. Then, using Unique Possibilities, one can fill in additional cells to get to: # More...
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Hi Steve,

Having a read a couple of your proofs (of the tough ones), I think that I can more or less follow your notation and logic (being a mathematician myself, that was not too difficult).

One thing I keep wondering about. How can you be sure that you always have found all the More...
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A number of questions are posed(implicitly and/or explicitly):
1) How can I be sure that I have found all the 'UP''s?
After solving a puzzle, but before writing a proof, I use a computer program like Simple Sudoku to insure that I did not miss any UP's.

2) Why bother posting More...
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Thanks for the informative and prompt reply. Yes, I do realize that the availabel space places very severe constraints. Also the traditional way of writing things has the benefit of having passed the test of history. The upshot being that some choices had to be made, and a More...
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While I'm at it I want to discuss a point about sudokus that has been bothering me. How do the regulars feel about the use of the fact that a well designed sudoku has a unique solution while solving it? I mean, I have encountered several times a situation, where I could make further deductions More...
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This does come up occasionally. What you're describing is a variation of 'unique rectangles', which uses the assumption that the sudoku has a unique solution to make eliminations. Different people have different opinions on it. I'm not above using it but I try to avoid it.
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Check out the comments in the tough sudoku from July 14, 2006 for a unique rectangles proof and some discussion.
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Using uniqueness of solution is widely accepted in the Sukoku community - but not by all.

If one is to use the quality, it should be identified as an axiom. Therefor, I have included Uniqueness of solution as one of the rules in my introductory blog of December 08.

One of the techniques derived from this rule, 'Unique Rectangles', will be examined in an upcoming blog.
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Thanks Clark & Steve,

It does sound like people have at least somewhat mixed feelings about the use of unique rectangles. I guess that when racing against the clock anything goes (particularly without knowing how the unknown adversaries feel about it :-).

However, I would prefer all More...
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I'm pretty new to Soduko, but just love it. I still have some questions. In the first box, second row I have 123 and a 13. In the last box, second row I have a 23. What is this an example of? and can I eliminate any numbers because of it? Yeah, I know, sounds simple enough...I've read a lot of the tips, but some of them are really confusing. Anyway, I would appreciate any help. Thanks, Rob
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Hi Robin!

You have a naked triple 123 in the second row. Even though 123 does not exist in all three cells, the requirement is that the three cells are limited to no more than the same three candidates each to form the naked triple.

You can eliminate 123 from all the cells in the More...
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Hey Steve! What is a 'QAUD' [see title] (I am hoping you just did not spell 'QUAD' correctly.)
I once worked for a man who headed a group which designed radar receiving equipment, yet he always misspelled 'receiver.' He said 'I am an engineer, we are notorious for poor spelling,' and my retort More...
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Thanks Jim!
A page with a blatant error, in the title, for this long!

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