- 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).
Consider:
- 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 .
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 .
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:
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.
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