Almost Locked Sets


The technique, Almost Locked Sets is a type of Advanced Forbidding Chain. If this is your first visit to my blog, WELCOME! You may find it useful to read some other blog pages first. Links to these can be found to the right, under Previous Entries. Almost Locked Sets are a powerful idea.

Some definitions

Naked N tuple
A set of N cells limited to no more than N candidates within a common container(s)
  1. N=1 would be a solved cell (naked single)
  2. N=2 would be a Naked Pair
  3. N=3 would be a Naked Triple
  4. N=4 would be a Naked Quad
  5. etc
Hidden N tuple
A set of N candidates limited to no more than N cells within a common container(s)
  1. N=1 would be a solved cell (hidden single)
  2. N=2 would be a Hidden Pair
  3. etc
N tuple
Naked N tuple or Hidden N tuple
Locked Set
N tuple
A set with M Degrees of Freedom
One of:
  1. A set of N cells limited to no more than N+M candidates within a common container(s)
  2. A set of N candidates limited to no more N+M than cells within a common container(s)
Almost Locked Set
A set with 1 degree of freedom
Almost Almost Locked set
A set with 2 degrees of freedom
Almost^M Locked Set (ALS(M))
A set with M degrees of freedom
These definitions are primarily provided so that if you visit other Sudoku sites, you can understand the terminology commonly used there.

Examples


Almost Locked Sets - One degree of freedom

Naked Almost Locked Set Examples

Hidden Almost Locked Set Example


Almost Almost Locked Set - Two degrees of freedom

Naked Almost Almost Locked Set Example

For the purposes of this page, one really only need consider Almost Locked Sets = Tuples with 1 degree of freedom.

The general idea

Suppose that S is:

  • A Naked Almost Locked Set (has one degree of freedom)
  • Contains the candidate r
Then one could say:
  • S is a Naked N tuple OR
  • S contains candidate r
If this is not immediately obvious, consider:
  • If S does not contain r, it is a naked n tuple
  • S must either contain r, or not.
We could write this as part of a forbidding chain:
  • (S is a naked ntuple) == (S contains r)

Simlarily, suppose that S is:

  • A Hidden Almost Locked set (has one degree of freedom>
  • contains the cell x
Then one could say:
  • S is a Hidden N tuple OR
  • S contains cell x
We could write this as part of a forbidding chain:
  • (S is a hidden ntuple) == (S contains x)
Hopefully the symmetry is clear.

It should now be fairly obvious that one can use an Almost Locked Set configuration within a forbidding chain.


Linking the idea

The idea of Almost Locked Sets is of little use unless it links with at least something else.

Clearly, since the idea of ALS is actually a strong link in the chain, it needs to link to the rest of the chain using weak links. Really, this is all one needs to know. However, in the common usage of this concept elsewhere, the protocol is to link one ALS to another ALS. This restriction is hardly necessary. Below are some examples of linking two ALS's together.


Typical ALS examples


Y wing as ALS

Y wing as ALS

The example above is also commonly described as a Y wing. One can, however, interpret this configuration as Two ALS's with a weak link between them.

  • a1=12 is an almost single with one degree of freedom
  • h1=29 and g2=19 is an almost pair with one degree of freedom
  • a1=2 -- h1=2 links the two almost locked sets
  • The 1's are therefor strong in one set or the other.
  • Therefor, a1=1 == g2=1 forbids abc2=1,gi1=1
As a forbidding chain, rewritten to reflect the ALS:
  • a1=1 == a1=2 -- h1=2 == {Pair 9, 19 at h1,g2 respectively}
    • proves a1=1 == g2=1, forbids abc2=1,gi1=1
  • This example is of little use if you understand a Y wing, but it is helpful in understanding how techniques that seem unrelated are in fact closely related, if not identical.


    Y wing style as ALS

    Y wing style as ALS

    I have treated the example above previously as a Y wing style. Clearly, a1=2 forbids h1=2. Therefor, a1=1 == {pair 19 at g2h1}. As a forbidding chain, one could write:

    • a1=1 == a1=2 -- h1=2 == {pair 19 at g2h1} forbids gi1==1
    As an ALS logic, one would say:
    • S1 = {a1=12}
    • S2 = {h1=129, g2=19}
    • Note that S1, S2 are both ALS with 1 degree of freedom
    • Note that 2's are weak across sets S1, S2
    • Therefor, {S1 contains 1} == {S2 contains 1}
    • Therefor, a1=1 == {h1g2contains 1} forbids gi1=1


    Almost Triple with Almost Pair

    Triple OR Pair

    Key: Red ~4 means 4 is excluded by this pattern at these locations

    Here we have:

  • Triple 12,124,124 at a2,ac3 respectively but for the possible 3 at a2
  • Pair 45 at h2i1 but for the possible 3 at h2
  • Since a2=3 and h2=3 cannot both occur, clearly:
    • {Triple 12,124,124 at a2,ac3 respectively} == {pair 45 at g2i1}
    • Forbids ghi3=4
  • As a forbidding chain:
    • {Triple 12,124 at a2,ac3} == a2=3 -- h2=3 =={Pair 45 at h2i1} forbids ghi3=4
    As ALS style logic:
    • Let S1 = {a2=123,a3=124,c3=124} - note ALS with one degree of freedom
    • Let S2 = {h2=345, i1=45} - note ALS with one degree of freedom
    • 3 is a restricted common candidate across the sets S1,S2
      • 4's are strong within the sets
      • All cells seen by the 4's all the 4's within S1,S2 are forbidden
      • Thus ghi3=3 is forbidden
    The forbidding chain style of expressing this idea is more concise than the alternatives.


    Almost Triple with Almost Hidden Pair

    Naked Triple OR Hidden Pair

    Key:

    • Black marks = native strong cells with candidates listed
    • * = wild card (could be any candidate(s)
    • Green ~15 = cells that, as a native puzzle condition, cannot contain 15. Therefor indicates strength in location for those candidates 15.
    • Red ~2 = cell proven not to be 2
    Here, we have:
    • Hidden pair 15 at g2,i1 but for the possible 1 at h3
    • Triple 34,234 at a3,ab2 respectively but for the possible 1 at a2
    • Since a3=1 and h3=1 cannot both occur at the same time,
      • {Hidden pair 15 at g2i1} == {Triple 34,234 at a3,ab2 respectively}
      • forbids g2=2
    As a forbidding chain:
    • {Hidden pair 15 at g2i1} == h3=1 -- a3=1 == {Triple 34,234 at a3,ab2} forbids g2=2
    If this one escapes you, then consider what if g2=2. Then pair 34 at ab2 would force a3=1, but g2=5 == i1=5 would force i1=5, but the strong set{g2=1,i1=1,h3=1} would force h3=1. Clearly, both a3=1 and h3=1 cannot occur. There is always an equivalent proof by contradiction available for every possible sudoku technique.


    Aligned Almost Triple with Almost Hidden Pair

    Naked Triple OR Hidden Pair Again

    The example above is almost precisely like the previous one, but the location of the Almost Hidden Pair 15 allows one to make an additional exclusion as illustrated.


    Almost Hidden Pair with Almost Hidden Triple

    Hidden Triple OR Hidden Pair

    The example above is a bit complex. It is probably better understood with a bit more chaining instead of using the Almost Hidden Triple, since we also have two Almost hidden Pairs. As a forbidding chain:

    • {Hidden pair 23 at gh2} == b2=2 -- b5=2 == {Hidden pair 12 at gi5} -- gi5=4 == h6=4
      • forbids h2=4



    Logical extensions of the concept

    If you consider the concept carefully, all strong sets on the puzzle grid are Almost Locked Strong Sets with some degree of freedom. For this reason, the concept easily generalizes beyond using only Almost n tuples, to also using, well, Almost Anything. Concepts that can be similarily analyzed include (but are not limited to)

  • Almost Fish (Xwings, Swordfish, Finned Swordfish, Jellyfish, etc)
  • Almost Almost Unique Rectangles
  • Almost coloring
  • Almost Locked Candidates
  • Almost Forbidding Chains
  • Of these, the last one is the most interesting


    An almost pair with an almost forbidding chain

    Almost Naked Pair with an almost forbidding chain

    If you are very clever, you may see the pattern above as An Almost Pair strongly linked to an Almost Hidden Pair. In this way, it is clearly also a subset of the general idea of Almost Locked Sets. Nevertheless, that expression of this elimination is fairly clumsy. I prefer to see it this way:

    • b1=1 == c2=1 -- c2=4 == a3=4 -- gi3=4 == {Pair 12 at gi3} -- ghi1=1


    An almost pair with an almost forbidding chain forming a wrap around forbidding chain

    Almost Naked pair in a closed loop

    key

    • Black marks are native strength in cells
    • * is a wild card
    • ~ means not
    • Green marks indicate, by exclusion, native strength in location
    • Red marks indicate items forbidden by the pattern
    The exclusions above are justified by the following "wrap around", or closed loop, forbidding chain
    • h2=1 == c2=1 -- c2=4 == a3=4 -- gi3=4 == {pair 12 at gi3}, but {pair 12 at gi3} -- h2=1, therefor forbids
      • c2=2356789
      • ghi,h3=1
      • defh3=4
      • abcdefh3=2 and ghi12=2
    There is much bang for the buck here. Because this is a wrap around chain, there are many equivalent ways to present the same idea. An instructive, albeit not very transparent one is :
    • g3=2 == {Y wing style wrap around chain at hc2=1, c2a3=4, g3=14} -- i3=14 == i3=2, but i3=2 -- g3=2, therefor forbids (same items listed in chain above)
    This previous representation illustrates the almost infinite flexibility of using forbidding chains, as it uses an Almost Locked Y Wing Style Chain and and Almost Almost Locked set.


    Almost Hidden Pair with an Almost Forbidding Chain

    Almost Hidden Pair Chained

    If you are clever, you may see the elimination above as an Almost Locked Hidden Pair used with an Almost Naked Pair. That is a bit clumsy, so I prefer:

    • {Hidden pair 19 at ac1} == d1=9 -- d4=9 == d4=2 -- a4=2 == a4=3 forbids a1=3


    Summary

    The concept of Almost Locked Sets is a very powerful one. The more generally one applies the concept, the more powerful it is. In its most general form, it is equivalent to forbidding chains, and probably can be used to justify every possible sudoku elimination. Nevertheless, its representation can be a bit unwieldy, so forbidding chains are probably a better language and logic system choice. However, since the concept allows one to easily take previously learned techniques and apply them at will, it is a valuable concept to understand fully.


    Some Practice Puzzles




    5 Comments
    Indicate which comments you would like to be able to see

    If there is a particular technique that anyone wants to suggest for the blog to treat soon, let me know. I appreciate the input!

     |  |
    Thank you, Steve. This explains so much - now for the practice! Btw, are there a couple of typos in the fc for Almost HP with Al.HT, or is it just me? Should it -- b5=2 ==(HP14 @ gi5)? or b5=2==(HP12 @ gi5)? And forbids g2=4 or h2=4? Cheers, and thanks again.
     |  |
    Thanks giblet!

    Indeed it should be HP12 @ gi5. And, the fc forbids h2=4, not g2=4. I will correct the page.

    I really appreciate when people inform me of typo's. Often times, I just see the eliminations and forget to carefully proofread what I wrote.

     |  |
    i fail to see any almost hidden triple in the example above (AHT .. AHP)
    05/May/10 3:42 AM
     |  |
    hi farpointer! I could have written:
    {Hidden pair 23 at gh2} == b2=2 -- b5=2 == (2)gi5 -- (2)ghi46 == (Hidden triple 124)at gi5,h6 with (4) limited to h6.

    It is a bit strained, but we do have (3) candidates (124) limited in one box to three cells: gi5, h6. That satisfies the definition More...
    05/May/10 3:53 AM
     |  |
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