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Strategies for Popular Number Puzzles

Hidden Candidates

Hidden Pairs

Looking for Hidden Pairs is a great way to open up the board. This approach can remove a cluster of candidates from two cells and leave behind simple pairs which are the building blocks of more complex elimination strategies.

Hidden Pair example
Hidden Pair example : From the Start
Looking at the top of this moderate puzzle, we see that 6 and 7 have been found in the first two boxes. Along with the 6 and 7 in column 7, this pins the placement of 6 and 7 in the third box to A8 and A9. It still appears that there are a great number of other candidates in A8 and A9, which is true up to a point. However, these extra candidates 'hide' the true values for these cells. We have deduced that 6 and 7 must go in A8 and A9 and therefore we can clear off all the alternatives. This doesn't mean we know which way round the 6 and 7 will go, but we can make 6 and 7 a Naked Pair in those cells and see where it leads us.
Three Hidden Pairs
Three Hidden Pairs : Load Example or : From the Start

This is a more interesting and complex set of Hidden Pairs. Three occur simultaneously. In the blue rectangle, [2,4] form a Pair on D3 and E3, clearing off 3, 5, 6 and 7. The red cells indicate two Hidden Pairs based on [3,7], which form a neat corner of three cells. [3,7] is unique to two cells in row E and two cells in column 7. The yellow highlighted cells can be removed.

Hidden Triples

We can extend Hidden Pairs to Hidden Triples or even Hidden Quads. A Triple will consist of three pairs of numbers lying in three cells in the same row, column or box, such as [4,8,9], [4,8,9] and [4,8,9]. However, in just the same manner as Naked Triples, we don't need exactly three pairs of numbers in three cells for the rules to apply. Only that in total there are three numbers remaining in three cells, so [4,8], [4,9] and [8,9] is equally valid. Hidden Triples will be disguised by other candidates on those cells, so we have to prise them out by ensuring the Triple applies to at least one unit.

Two Hidden Triples
Two Hidden Triples : Load Example or : From the Start

This tough puzzle has two Hidden Triples: the first, marked in red, is in row A. Cell A4 contains [2,5,6], A7 has [2,6] and cell A9 contains [2,5]. These three cells are the last remaining cells in row A which can contain 2, 5 and 6, so those numbers must go in those cells. Therefore we can remove the other candidates.

Now that we've removed those candidates from the red cells, we can see in column 9 that [4,7,8] is unique to cells B9, C9 and F9. By the same logic we can clear off other candidates in those cells.

(The solver will not choose the second example as Naked Triples get there first)
Hidden Quad
Hidden Quad : Load Example or : From the Start

Hidden Quads

Here is the one example of a Hidden Quad I found in a set of 18,000 Sudoku puzzles. Four numbers [3/4/5/7] on four cells are hidden by just two 6s in column 7. Barely qualifies as 'hidden', but it is legitimate. Note how none of the cells need to have all four numbers, as long as only four cells contain all four numbers and are intermingled.

Hidden Quads almost always only occur in rows, columns and boxes where there are no clues or solved cells, so you can be forgiven for skipping them outside those circumstances.
Hidden Quad
Hidden Quad : Load Example or : From the Start

Klaus Brenner in Germany has found a number of excellent Hidden Quads, and I include one here to show they do exist.

The Hidden Quad is {1,4,6,9} in Box 5 and exists only in the four cells [D4,D6,F4,F6]. Therefore other candidates (red text on yellow background) can be removed.

This very special puzzle also produces a perfectly formed Empty Rectangle later on.
We don't consider higher orders of Hidden candidates because there are only 9 cells in a unit. So if we were to suppose a "Hidden Quin" with five candidates there would automatically be a complementary Hidden Quad since 5 + 4 = 9. Same point arises with Naked sets. It may be viable to look for such beasts in 12x12 or 16x16 Sudokus.



... by: Michael

Thursday 7-Sep-2023
I was looking through this page and the discussion in order to gain some insight into finding hidden triples. I came across the comment by Peter Heichellheim and response from Andrew below on 23 September, 2013.

It seems like a very useful rule, but tried to validate it by working backwards by starting with a unit with a very simple 1,2 - 2,3 -1,3 triple in the unit. I then chose 3 cells where I arbitrarily assigned a solved number (other than 1, 2, or 3). However, I was not able to validate the rule. Not sure what I am doing wrong, so would appreciate some logical insight to why this rule works.

"There is a clue which will help someone in spotting hidden triples. If there is a hidden triple it must be in a unit with two or less solved squares.
Andrew Stuart writes:

This is a good insight."

... by: Ymiros

Thursday 22-Jun-2023
Last few sentences isn't a hidden quint automatically accompanied by a naked quad rather than a "complementary Hidden Quad"?
Take your final example for example, the hidden quad is 1469 and the complementary naked quint is 23578 in the cross shape section of the middle box.

Quite neat explanations though, thank you very much for that!

... by: tofubob

Thursday 16-Apr-2020
In your last example of a Hidden Quad {1,4,6,9}, could you not have picked {3,5,7,8} just as well?
Andrew Stuart writes:
Hurts my head to think about it so going with 'first Quad the solver found'
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... by: Leo J

Sunday 8-Dec-2019
In the first Hidden Quad puzzle:
- why no "4" in cell J7? (how did you know it CAN'T be a 4?)
- why no "2" in cell G7?

thanks/i understand if you don't have time to answer this.

Andrew Stuart writes:
Previous eliminations. This is not an ideal puzzle to show an example from since there are many complex eliminations before the Hidden Quad is shown. I will try and find a better example and replace it.
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... by: Nathan

Tuesday 15-Oct-2019
I have a question. So, I was reading through all of your startegies, and I started finding 12x12, 16x16, even monstorous 20x20 and 25x25 sodukus online. I just want to know if the principles for any of your strategies change when puzzles get larger and/or more complex.
Andrew Stuart writes:
Nothing special or unique pops out with larger Sudoku. Just the alignment of boxes may invalidate certain strategies that reply on certain alignments or overlaps. Personally I find them a chore as they offer nothing new but take much longer.
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... by: alan g

Monday 26-Aug-2019
is there such a thing as an ambiguous sudoku puzzle?
I recently encountered an "easy" puzzle but it proved to be the hardest one I've met.
I got as far as I could with it but couldn't make any further progress until I found this page which told me about hidden pairs etc..
I eventually solved it but the answer I obtained was slightly different to the one given in the paper.
however every row column and mini grid in my solution, each contained the numbers 1-9, which is surely the definition of the correct solution.
the trouble with this particular puzzle is that there are many ways to apply the hidden pairs to, and I think the solution depends on where you start
Andrew Stuart writes:
Might be faulty and have several solutions. Always use Solution Count to check first.
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... by: Liz

Tuesday 5-Mar-2019
If "hidden" means that each member of a set of n candidates only occurs in a set of n squares in a unit, then if there are N already solved squares in that unit, does it not follow that the remaining 9 minus N minus n candidates must form a *naked* set? Since, otherwise, one of the n candidates would have to occur in at least one of those remaining squares?

Is it not generally easier to find a naked set than a hidden one, even if the naked set is a 5, 6, or even 7-tuple?

In your first example above, I would have noticed first that 4,7,8,9 form a naked quad, making a triple out of 2,5,6. Even in your last example, 2,3,5,7,8 form a fairly obvious naked quint.

Because of this I'm having trouble understanding why "hidden candidates" is even discussed as a separate strategy.

Or are there cases where a hidden set does not have a complementary naked set? (which depends, perhaps, on exactly how "hidden" is defined)

... by: pyorokun7

Friday 24-Nov-2017
Never mind, it does detect it, but only after elminating a lot of candidates in the column with several advanced techniques.

Also, by the time it detects it, one of the 6 is already eliminated

Hidden Quad 3/4/5/7 in Col 7, on cells [E7,F7,G7,H7]
- removes 6 from H7

(the other 6 was removed by the APE that eliminated the 4 in J7, creating the Hidden Quad itself)

So probably it should be updated, or maybe should use another example.

By the way, is there a reason why the last row is J, instead of I?
Andrew Stuart writes:
Because I looks too much like 1 and this small changes reduces potential confusion.
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... by: pyorokun7

Friday 24-Nov-2017

Your solver (I guess after an update) no longer detects this Hidden Quad in Col 7; after the Naked Pair in Box 3 of 3/7, it jumps to X-Wing

... by: Hamza Zaidi

Saturday 1-Jul-2017
I don't get the last sentence. What is a complimentary hidden quad?
Andrew Stuart writes:
If there was such a thing as a 5-number Quin it would have a "complementary" Quad.
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... by: KDot93

Sunday 13-Mar-2016
***In response to YBB (below)***: I believe a "hidden quad" is just a "naked quad" with an interfering number - in the case of the example, the possibility of a 6 in cells G7 and H7 made that quad into a "hidden quad". However for your question, since there is no interfering number, A7,B7,C7 and C9 form a "naked quad" instead (as well as B9 and C8 forming a "naked pair").

PS: I'm new to advanced sudoku-ing/this website, so if that's an incorrect explanation/there is a direct "reply" function, please let me know! (somehow...). Or perhaps if Andrew could make an "answer/reply" function so he doesn't have to respond to all the comments, that would be great!

PPS: Really awesome website!

... by: YBB

Thursday 25-Feb-2016
With reference to first example on Hidden Quads, why is 1,6,8,9 in the top right box not a hidden quad?

... by: Jwamyr

Friday 17-Jul-2015
I have made an observation that any time a hidden pair or triple or quad is found, you can also find a naked N in the remaining cells of thathe set. For example, a naked set of 5 can be found in d5, e4, e5, e6, and f5 in the example given for the hidden quad. I have not proven that this is true always but I believe that it is.

... by: Didi

Saturday 23-Aug-2014
Sorry, I have to correct my last comment a bit:
If you can use a single or pair for pointing, it's regardless of whether it's hidden or naked. The thing that I wanted to point out is that if you've found "some stuff" inside a block ("claiming", "pointing", "naked/hidden pair"), then you always should keep your focus where you currently are, to watch out for more of that "stuff" that could be found now, either beeing human, or a computer solving the riddle.
P.S.: Your site here is brilliant, thank's a lot!

... by: Didi

Friday 22-Aug-2014
I think it should be mentioned that a hidden pair leads always, after processing, to a naked pair. Although this is a platitude, there's reason for being aware of this change. In some cases the resulting naked pair can be used as a pointing pair.

The fact is also interesting to programmers: If you want efficient computer code, this check can be done immediately and more effortlessly, rather than checking again with an extra-algorithm.
Andrew Stuart writes:
A true fact, very worth mentioning
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... by: Doc

Thursday 17-Apr-2014
In your hidden pair example in the blue boxes what determines it is 2,4 and not 2,6 and in the red boxes why 3,7 not 3,9.
Andrew Stuart writes:
There is a 6 in B3, so a 2/6 pair can't form in the column.
Likewise, there are other 9's in column 7 so only 3/7 makes a pair in that column
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... by: Peter Heichelheim

Monday 23-Sep-2013
There is a clue which will help someone in spotting hidden triples. If there is a hidden triple it must be in a unit with two or less solved squares.
Andrew Stuart writes:
This is a good insight.
curious replies:Thursday 12-Jan-2017
@Peter Heichelheim
>If there is a hidden triple it must be in a unit with two or less solved squares.

Can anyone explain the reasoning behind this? I feel like it's less an inherent truth and more along the lines of:

"If a unit has >2 solved squares then the solver probably already indirectly eliminated the hidden triple by using some other strategy elsewhere in that unit".

I REALLY want to understand this insight.
atn173 replies:Wednesday 6-Dec-2023
The insight by Peter can be explained as follows.
Suppose that there are 3 solved squares, say, 1,2,3, and that there is a hidden triple of 4,5,6, i.e. 3 squares that containing 4,5,6 and at least one of 7,8,9. Look at the other 3 squares. They must be naked triples containing 7,8,9, which would be much easier to spot and would have eliminated 7,8,9 from the original 3 squares in the first place.
So the insight is that in order for a unit to contain a true hidden triple (meaning no naked triple in this unit), the unit must contain at most 2 solved squares.
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... by: Y.Sato

Monday 9-Sep-2013
Your example of sudoku is very interesting, but the logic to find a hidden triple is hard to me.

In the example of Hidden Triples on this page,
I found two candidates of Hidden Triples in column 9 according to my rule.

A: { {4, 7, 8}, {4, 7, 8}, {4, 8} }

B: { {4, 8, 9}, {4, 8, 9}, {4, 8} }

Your solution is the candidate A.
Why is the candidate B not a solution ?
Would you please teach me the reason ?

Best regards,
Yoshihiro Sato
Andrew Stuart writes:
Maybe a poor example, since the second Hidden Triple cannot work until the first Hidden Triple has removed the candidates. Since this example was published I've stopped these sorts of "knock-on" effects from appearing in the solver
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... by: tu_79

Wednesday 20-Mar-2013
I must agree with Chuck. It is really difficult for a human to detect hidden triples, and otrher strategies like simple chains are more easy to spot. Therefore, I would be great if we could unselect Hidden triples/quads from your solver.

... by: ad.joe

Monday 28-Jan-2013
Hello Andrew, this time a naming suggestion to differentiate between basic and advanced techniques:


As can easy be seen in example 2 of the Hidden Pairs, it only deals with column 7!

So it's like this:
Basic strategies deal with only on row, one column OR one box, therefore are working ONEDIMENSIONAL! While advanced techniques/strategies are working moredimensional: in more than one row AND column AND work.

And I've never read such a sentence "anywhere", what do YOU say?
Besides a hidden subset is easier to be understood (for most people) as an x-wing, it can still hard to be found, becaus one has to look up the occurences (2 for instance) of the numbers in this one dimension.
Andrew Stuart writes:
It's only one-dimensional in a loose, not very practical sense, in that one row or one column or one box may be involved. But in a box, for example, the three cells could be spread out in different rows and columns, spoiling the "one-dimensionality"
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... by: m&mjabb

Tuesday 1-Jan-2013
hidden triples example - this wouldn't be a triple but in row 9 why couldn't the 4,7,8,9....4,7,8....4789....48 be used as quad? is it because of the 9 in row J?
(beginner sudoku player)
Andrew Stuart writes:
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... by: amarabavani

Tuesday 18-Sep-2012
Fantastic page. Your explanations are simple and the working user-friendly. keep up the good work and good luck.

... by: adivandhya

Tuesday 10-Jan-2012
i tried to download the website so that i can read it while im offline, nw it says i am a bot..wht should i do to get the access back?
Andrew Stuart writes:
Sorry you got caught by my anti-hacking, bot-kicking scripts but I have to have some means of detecting non-humans. If I knew your IP address I'd be happy to unblock you. But you may not be able to read this message...
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... by: Ritesh

Saturday 12-Nov-2011
Firstly , thanks 4 Sharing the site ...
Give me more examples to understand hidden Quads

... by: Chuck Watson

Sunday 11-Sep-2011
Great site! I have learned to solve much harder puzzles by studying this site.

What I wanted to point out is your last example of a hidden quad actually also shows a naked quintuple. To me the naked quintuple is easier to see.
Andrew Stuart writes:
If the puzzle is 9 by 9 then a quintuple will automatically be the mirror of a quad. Which is why we don't extend this series beyond 4. But i agree it is often possible to identify the mirror side of a set rather than the set itself.
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... by: Joseph Badillo

Wednesday 20-Jul-2011
Very nice site, very helpfull.
Question regarding making a mistake:
Your almost to the end and you find out that a 6 and a 9 does have been repeated. Are there short cuts you can take to find out where you made a mistake other than starting over?

... by: Walsh

Saturday 19-Feb-2011
From that example you can see that the 4 has to be part of the hidden triple since it has to be in one of those two squares. The three and one cannot be used because it has possiblilities in other places in that block. A Hidden triple will have a sequence of numbers that are in only three boxes in a row, column, or block.

... by: Jim R

Thursday 17-Feb-2011
Bill: the idea of the Naked Triple is that the three numbers (4/8/9 in this example) only appear in exactly three squares. You can find some combination of 1/8/9 in all seven unsolved squares. 1/3/9 also cannot be the triple because there is a 1, 3 or 9 in all seven unsolved squares.

... by: Bill

Monday 31-Jan-2011
I am looking at the hidden triple example. Your results is 4/8/9; however, why can't the hidden triple be 1/8/9 or 1/3/9?

... by: David Harkness

Saturday 30-Oct-2010
Lloyd, this strategy allows you to eliminate the other numbers from the cells containing the hidden pair/triple/etc. It does not allow you to remove the numbers in the pair/triple from other cells.

The key is that the three numbers involved in the triple must occur in only three cells. In your example, 124 is not a triple because 2 is a possible in 5 cells. Each number must be a possible in at most 3 cells (2 for a hidden pair, 4 for a hidden quad, etc).

Naked Pairs is where you eliminate the numbers in the pair from other cells.

Andrew, the section on triples mentions "three pairs" of numbers, but often one of the cells will contain all three numbers. Is "pairs" here simply a copy/paste error from the section above, or does it have some other significance? I understand that sometimes the numbers appear in pairs in 3 cells, but that's not necessary.

Also, the first paragraph has an extra word: "hidden in in the squares".

... by: Lloyd Pape

Saturday 5-Jun-2010
Hi, I get confused. I have a row: 24,14,1245,257,2567,3,1267,8,9 ,..I would like to group cols 1,2 and 3 with a tripple, 24,14,124,delete the 5 in col 3,...then, delete the 2 incol4, the2 in col 5 and the 12 in col 7, that I end up with a tripple ,1,2,4, another tripple 5,6,7 and 3,8,9,...does this work,..if so , are the tripples naked or hidden,...thanks for your info to date,....regards,...lloyd

... by: Sarah

Thursday 6-May-2010
Hi Andrew,
Great site. I'm terrible at spotting hidden triples. Any tips, hints on becoming better at this?

... by: csvidyasagar

Monday 8-Mar-2010
You have given an excellent example of Hidden Pairs, Hidden Triples and Hidden Quads. But the example you quoted of Mr Klaus Brenner of Hidden Quad, I suppose is NOT Correct. For hidden pairs, the digits must appear only in two cells like you correctly shown as 58 in your example. The digits 5, 8 do not appear in any other cell of the box. In Hidden Triples of digits 4, 8, 9 lie in only three cells and none of them appear in any other cell in the box. Similarly in Hidden Quads of your example, the digits 3,5,6,7 appear only in four cells in the Box and do not appear in any other cell of Column 8. But it is not so in the example cited of Mr Klaus Brenner. May be I am wrong or my understanindg of Hiddens is wrong. So can you explain how Hidden Quads are there in the example of which you have high lighted some cells.
Andrew Stuart writes:
The Hidden Quad is {1,6,7,8} and exists only in those four cells. Therefore other canidates (yellow/red text) can be removed.
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... by: robin smith

Monday 15-Feb-2010
Hi, What a great site . it's so helpful when I really get stuck with a puzzle. One minor point . . would it not be possible to have the original numbers in the puzzle in a different colour so it is more easy to spot the ones that get added as the solution unfolds.

Just a thought.

But it's stil a great site.

Robin Smith
Andrew Stuart writes:
Orginal colours (clues) are red, solved cells are blue. There may be the odd diagram which I have snapped where this is not the case and i'll be replacing when I spot those
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... by: Maurice Popple

Saturday 21-Nov-2009
I'm new to Sudoku and am quite bewildered when I manage to solve one.
Questions: (1)How many unique (ie. one solution) puzzles exist in the 9X9 Sudoku matrix (I tried to calculate and got a brute of a binominal I could not resolve)? (2)What is the minimum number of given entries at the start to uniquely define a puzzle?
Many thanks, Maurice
Andrew Stuart writes:
Number of ways to fill a 9x9 grid is 6,670,903,752,021,072,936,960 - so you have to multiply that with all possible blanks. 17 is the minimum number of clues.
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... by: Chuck Bruno

Tuesday 1-Sep-2009
Hello Andrew,

I have written several times but I must repeat "This is a great site". That being said, I have a question:

How do you determine what the difficulty level should be for a given stratagy?

I personally find that Pointing Pairs, Box/Line Reduction, X-Wing, and Unique Rectangles, are much easier to spot than Hidden Triples. When I get stumped on a Sudoku, I import it into your Solver, uncheck stratagies I don't usually look for, and step through it. In most cases, I find that I simply overlooked something silly. In some cases however, the Solver finds a hidden double or triple. Because these can't be turned off like the more difficult stratagies, I can't force the Solver around it to see if one of the other stratagies that I normally use, would allow the puzzle to be solved without using the naked double or triple.

In summary, I would like to be able to selectively turn off the "easier" stratagies just as can be done with the tougher ones.

Thanks for your time,

Chuck Bruno
Andrew Stuart writes:
Sorry for the late reply. All valid points, I might have a go at that - but trying not to over complicate an already complex interface.
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... by: Roger

Friday 8-May-2009
Don't understand "from a library of 18,000"
Andrew Stuart writes:
A stock, or list of Sudoku puzzle definitions. I call such lists "libraries"
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