Solve Sudoku using Naked Pairs Triples or Quads

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 .

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.

Next, we extend this technique using Hidden Pairs, Triples and Quads

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

Steve  From Ohio    Supporting Member
Check out my page
Hi!
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 'The night before Christmas' poem tumbled into my head. I thought that I might finish it cleverly, but the task proved a bit too much for me. So, I have cobbled together a completed version of the parody, but probably will prefer to modify it greatly some day - or perhaps just erase the entire endeavor from memory.

Given that caveat - please forgive the following post. Hopefully, at least some of you find it at least fun.
Steve  From Ohio    Supporting Member
Check out my page
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 exclusion were still my plight.
Looking for locked candidates, found a few more,
But settled carefully they were, no fruit they bore.

When upon my 'puter screen splashed sudden chatter,
I spun in my chair to examine the matter.
Upon my keyboard I flurried a few fast strokes,
Perused pairs of pages and placed plenty pokes.

Cascading on my screen 'twas trickles of light,
Peeling off possibilities ever so slight.
When, what as my blurry sockets blankly stared,
Were paths a'plenty the puzzle to be pared.

Presented by a solver, skillfully so sage,
I knew in a moment this must be a good page.
From ev'ry grid corner techniques were presented,
Each purely posed, their power playfully rented.

'Now hi'en Pair, now, Triple! now, Swordfish and Xwing!
On, Colors! On ALS! On Chains and Ywing!
To scan the silly squares! to simplify cells all!
Now tame the posted puzzle! Make the numbers fall!'

As ideas complex 'fore the taming mind comply,
When they meet with a conundrum, vexing on high.
So over the puzzle placements, the tips they blew,
With myriad more methods, and gb there too!

And then, with but an inkling, I felt in my mind,
The clawing at candidates almost with rhyme.
As I pondered and puzzled and played with them all,
Upon my stalled screen Solver showed how numbers could fall.

Proof posted in jargon, thorough start to the end,
All obscured with language, tricky to comprehend.
A myriad of methods mystified my mind,
The sense of solving salvos, was hard to unwind.

The marks - how they promised! the language so compact!
With tactics twisted tersely, like matters of fact!
The yearning I had, the slick cipher to untwist.
If only I could understand that lousy list!

The Solver, he spoke, with words far, far too many,
But sense they did make, slowly taming the frenzy.
His staggered style, it presented somewhat stiffly,
But through it I waded, finding it almost nifty.

The numbers he loosened, slaughtered with zest.
Help such as this, placed that darned puzzle to rest.
Now through the panicky roar spun within my head,
Soon gave me to know, no puzzle should I dread.

Within that deep well, filled with lucid new leads,
What once was so mystic, now seemed merely a tease.
All those ideas, so many and oh full of disdain,
Were clearly to me now, simply but a chain.

The forbidding chains sat there all plainly inked,
A matrix of techniques commonly linked.
But I heard it exclaimed, as the cells cascaded to light
'Happy Sudoku'ing to all, and to all a great night!'
Aschli  From Ohio
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. haha...

Naw, with all honesty, I enjoyed reading this post. The poem was cute and really creative... you are quite the poet! Look forward to reading more! ;-)
Stella  From Saratoga, NY
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.
   bluey  From Port Kembla    Supporting Member
Check out my page
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!!!
Steve  From Ohio    Supporting Member
Check out my page
Hi Stella!
I usually write:

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

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: # cells filled in.

Others generally shorten the writing to:
Start at ## filled. (etc.)
The meaning is exactly the same, though.
To Steve and other provers  From Finland
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 UPs? Of course, even I can find them eventually, and they are easier than the forbidding chains for sure.
However, I sometimes feel that some hidden/blind
groups are easier to spot than some UPs :) May be that is because my sudoku solving thinking is not at all structured, but consists simply looking for the accumulated tricks brought into being by evolution.

Anyway, is it so that regular provers just use the
number of UPs as a sort of mile post / checkpoint
measuring the overall progress of the proof??

I once tried my hand at writing down a proof, and among the very constructive feedback I found the suggestion to add the number of filled UPs somewhat amusing. To a sudoku solver at my level it surely is more helpful to list the UP positions
deduced at that points (and more likely than not needed in the steps that follow).

Of course, it would be impractical to write down everything, and the simple steps can be 'left to the reader'. I just find the UP count a somewhat uninformative way of communicating a solution.
I mean, it's not like I count the UPs, when I solve a puzzle.

Granted, there's a difference between a solution
and a proof. Most of the time I get stuck at the
tough level, so I have looked at your proofs for
clues for solving (as a learning process).
Apparently the level supporting members the proofs are used for purposes other than communicating a compressed solution???

Cheers,

Jyrki
Steve  From Ohio    Supporting Member
Check out my page
Hi!
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 UP's?
Since it is impractical to post the exact puzzle stage that one has when one enacts a particular step, it is required that one uniquely define what that state is in some other way. When I post a proof, if you do no more than the precise steps listed, the puzzle will solve - uniquely. 'UP' is completely devoid of ambiguity, and therefor a perfect way to get from one puzzle state to another.

Every 'UP' is implicitly a group of candidate eliminations, and the goal of a proof is to state each elimination required, and to justify same.

3) Are n-tuples easier to see than UP's?
Of course difficulty is arbitrary, but for me, an n-tuple is often easier to find than an UP. The bag of tricks does not come with a required order for solving a puzzle. There is some logic, though - to a partial ordering of the bag of tricks when posting a proof.

4) Why not, instead of the 'UP' count, list the cells solved?
Convention. That is indeed a great question. I can only answer that those who posted proofs before me did it this way to probably save typing. Especially since at that time post length was limited much more. To list all the cell solvings would generally have pushed a proof into three posts. Perhaps now, this convention needs to be revisited.
jyrki  From Finland
Steve,

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 convenience probably quickly became a convention.

I really think that may be this is my personal problem. When a friend first referred me to sudoku.com.au I knew about hidden/blind groups (though I didn't know what they were called), but nothing about forbidding chains (but the said friend gave a hand-waving explanation of the thinking involved (= a longish proof by contradiction, often involving only a single digit). I still have trouble spotting them - especially, as I don't want to use markers.
(Never got used to them working on David Bodycombe's puzzles ;-)

Cheers,

Jyrki
jyrki  From Finland
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 along the lines of: if a certain digit
appeared in this set of cells, then there would be at least two distinct solutions to this sudoku. Ergo, it cannot be there.

The simplest such a scenario is as follows
('?' denotes an empty cell, '*' is a filled
cell):

? | ? | *
========== <-- a border between two 3x3 blocks
? | ? | ?1

assume that from the 3x3 block on top of the horizontal divide only digits 'a' and 'b',
and from the at the bottom digits 'a', 'b'
and 'c' are missing. Then one can conclude that
the cell marked with ?1 cannot be a 'c'.
For otherwise there would be two distinct ways
of putting the 'a':s and 'b':s into the remaining
four cells, and no further information from elsewhere in the puzzle to distinguish between the two.

I feel hesitant to use this kind of logic when
solving a puzzle. Obviously it is a no-no, when
writing down a proof. Would I use it in a race?
With money/beer/honor at stake? Perhaps?

Mind you, this is note as rare as it might sound.
OTOH my personal statistics on that may be skewed
for having played a lot of 16x16 sudokus (that didn't require using forbidding chains - spotting UPs and blind/hidden groups is trouble enough). Arguably there are more opportunities for this to happen in a 16 by 16.

Cheers,

Jyrki
Clark  From Michigan
Jyrki,

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.
Clark  From Michigan
Jyrki,

Check out the comments in the tough sudoku from July 14, 2006 for a unique rectangles proof and some discussion.
Steve  From Ohio    Supporting Member
Check out my page

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.
jyrki  From Finland
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 the sudokus that are
presented as puzzles to also have a humanly followable solution that doesn't need this trick.
(so that I can pretend that uniqueness is not a rule of sudoku per se - just an adopted convention
to avoid having to accept several solutions :-)

I discussed this with an Irish colleague (we had to find a way to pass the time on the train from Dublin to Cork). It turned out that he had set up a computer search (for uniquely solvable sudokus that have the smallest possible number of clues)
based on the heuristics that if you have a certain
number of rectangles with this symmetry possibility, then one of the cells of that rectangle has to be a clue. Thus you could cut down search time, by studying solved sudokus with a suitable number of such rectangles. He hasn't reported any success to me, but don't know if this due to bad heuristics or simply because the sought after minimum has already been attained.

Cheers,

Jyrki
Robin  From Louisiana
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
Steve  From Ohio    Supporting Member
Check out my page
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 second row outside of these three cells. (the other six cells). Additionally, since the 1's in the triple are also limited to two cells in the first box, you can also eliminate 1's from the 7 cells in that box that are not part of the triple.

I am sorry that some of the tips are really confusing. If you do not mind, I would love to hear about that. I am always interested in improving my presentation. You can email me at:
solidsudoku@yahoo.com with suggestions or explanations, etc.
Sir Jim  From Redpenland
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 was 'You are head of the radar receiving group, the least you could do is learn to spell 'receiver.'
He never got it wrong again.
My point is that poor spelling can distract the reader from 'receiving' the lesson.
Sincerely, The Spelling Nazi
Steve  From Ohio    Supporting Member
Check out my page
Thanks Jim!
WOW!
A page with a blatant error, in the title, for this long!

Please Log in to post a comment.

Not a member? Joining is quick and free. As a member you get heaps of benefits.

Join Now Login