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# Naked Candidates

'Naked' in this context refers to all the remaining possible candidates on a cell which are going to be used in a strategy. The simplest such situation is a Naked Single - or the last remaining candidate on a cell. Generally speaking, if you are making notes on a Sudoku board you have reached a point where simple scanning of the rows, columns and boxes has brought you no further solutions. But you will be finding plenty of Singles on the easier puzzles, and hopefully not too few on the hardest ones.

A Naked Single is exactly equivalent to saying "Ah Ha! Looking at that cell, I can see every other number either in the same box, the same row or the same column, so it's the only number that can fit."

Hidden candidates, mentioned below with regard to Pairs and so on, also have a Hidden Single equivalent. It occurs when you find a cell with lots of possible candidates, but you reason "well, X can't go anywhere else in either the row, column or box, so it must go here."

# Naked Pairs

A Naked Pair (also known as a Conjugate Pair) is a set of two candidate numbers sited in two cells that belong to at least one unit in common. That is, they reside in the same row, column or box.

It is clear that the solution will contain those values in those two cells, and all other candidates with those numbers can be removed from whatever unit(s) they have in common.

In this example, several Naked Pairs are available and I have highlighted two. In red in row A, cells A2 and A3 both contain 1 and 6. We don't know which way round the 1 and the 6 will eventually be - we will find out later as we finish the puzzle - but it means we can remove all other 1s and 6s in the row. The solver has highlighted these candidates in yellow. But A2 and A3 are also in the same box, so we can clear off the 1 in C1 as well.

The [6,7] in row C is also a Naked Pair. It is aligned just in the row, but it removes three other candidate 6s and 7s in the row. Combining both Naked Pairs, we get a solved cell of 8 in C1.

There are other Naked Pairs at this point. You can identify them yourself or load the puzzle up in the solver to see them.
Just to show that pairs don't have to be aligned on a row or column, in this group of pairs we have a [4,7] pair on H2 and J1, which removes some 7s in the same box. Two other Naked Pairs eliminate further candidates at this stage.

# Naked Triples

A Naked Triple is slightly more complicated because it does not always imply three numbers each in three cells.

Any group of three cells in the same unit that contain IN TOTAL three candidates is a Naked Triple.
Each cell can have two or three numbers, as long as in combination all three cells have only three numbers.
When this happens, the three candidates can be removed from all other cells in the same unit.

The combinations of candidates for a Naked Triple will be one of the following:

(123) (123) (123) - {3/3/3} (in terms of candidates per cell)
(123) (123) (12) - {3/3/2} (or some combination thereof)
(123) (12) (23) - {3/2/2/}
(12) (23) (13) - {2/2/2}

The last case is interesting and the advanced strategy Y-Wing uses this formation.

This first example is as straightforward as it gets. In row E, centre box, are the cells E4, E5 and E6 containing [5,8,9], [5,8] and [5,9] respectively. In total, those three cells contain [5,8,9], so we have fixed those numbers in those cells - just not which way round they will be. This allows us to remove those numbers from the rest of the unit the Triple is aligned on - namely the row.
We have two Naked Triples at the same time on this board, in columns 1 and 9. There is no trickery in these Triples because the cells that form the triples are the last three unsolved cells in those columns - so they are bound to contain the three remaining values. Given that fact, we can clear out those values from each box containing a Naked Triple (and only the box, since there is nothing to clear off in the columns). But the manoeuvre nets us a great deal of candidates and we get a solution of 9 in F8.

In terms of the candidates per cell, the column 1 triple is a {2/2/3} formation (reading down) and the second, column 9, is {3/2/3}.

A Naked Quad is rarer, especially in its full form, but is still useful if it can be spotted. The same logic from Naked Triples applies, but the reason it is so rare is because if a Quad is present, the remaining cells are more likely to be a Triple or Pair and the solver will highlight those first.

Well, I can't find an example in my 2012 stock, so I'm going to use the one found by Pieter from Australia. It's a cluster of cells in box 1. A1, B1, B2 and C1 collectively contain [1,5,6,8], so those numbers must occupy those cells. That allows us to remove the yellow highlighted candidates.
We don't consider higher orders of Naked candidates because there are only 9 cells in a unit. So if we were to suppose a "Naked Quin" with five candidates there would automatically be a complementary Quad since 5 + 4 = 9. Same point arises with Hidden sets, but it is worth noting that the Naked complement will be Hidden and the Hidden complement will be Naked. It may be viable to look for such beasts in 12x12 or 16x16 Sudokus.

 Go back to Getting Started Continue to Hidden Candidates

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## ... by: John Short

I have just come on this solver after doing Suduko for a number of years. I am thinking though that I must have been doing pretty basic level puzzles. For example I tended to be able to solfve without filling squares with all the options/ Generally did by inspection and a bit of jotting possibles.
Would you say therefore that it is unlikely to be able to complete the tougher puzzles without the solver at your elbow?
before using the solver I used to come unstuck occasionally and only find out when the last few steps remained. Then it was almost impossible to backtrack. The solver allows partial progress to be checked which is great. Very impressive piece of work
Andrew Stuart writes:

Hi John.
Depends. Most newspapers print relatively easy puzzles. Few require candidates everywhere. The solver is aimed at the tougher end and it's a long tail.

## ... by: George

I'm really having a hard time wrapping my head around these quads. In your example, you show 1 as a digit of the quad group, even though 1 exists in A2, A6, and H1. What makes the 1's in A1, B1&2, and C1 different from those? Same with 5. They are all over the place col's. 2, 3,and 4.
Andrew Stuart writes:

True but the four cells A1, B1, B2 and C1 contain in total just four different numbers, that's what makes them special. We need four in four to make a quad and it's the only quad pattern in that box.

## ... by: Janet

I notice that in examples of Sudoku solving the candidates are placed in certain positions within the squares. Please explain how to do this to enable easier solution making. Thank you.

Janet D.

## ... by: Ing

The complementary (not "complimentary") of naked candidates is hidden candidates, not another naked candidate. So you won't need to look for "naked quin" if you do look for "hidden quads". Same for the "hidden quins" vs "naked quads".
Andrew Stuart writes:

Appreciate the spelling correction, thanks. Complementary is descriptive of the naked/hidden relationship but can be used for other relationships.

## ... by: DanG

Hi,
Thank you for the solver,
I don't fully understand all the strategies, but i'm working on it.
Although i like most of the examples shown,
The examples for the Naked Triples are not very useful.
Because the cells are align and it automatically exclude candidates.
I don't call that a strategy. It is more simple elimination.
We need to find examples where the cells are NOT align.
Have a nice day.

## ... by: Liz

By what rule must a naked quin always have a complementary *naked* quad (in a unit with no solved cells)? Could it not be hidden instead?

I think the general rule is that in a unit with N solved cells, a naked n-tuple always has a complementary 9-minus-N-minus-n-tuple, but it could be naked or hidden. In your example above, with one solved cell, the 1568 naked quad reveals a 2347 hidden quad.

I have a conjecture that is sort of the converse: that any *hidden* tuple will always have a complementary *naked* tuple. I'm horrible at spotting hidden tuples unless I spot a complementary naked tuple. But although I've never seen an exception, logically I can't rule out the possibility that the complementary tuple might be hidden. Maybe I'm missing something obvious.

## ... by: David Ullman-Dougherty

Your tutorial on naked triples does not have an example of a naked triple in a box but not aligned on a row or column. I found this while using your solver. The Sodiku is from page 177 of Will Shortz presents Killer Sudoku.

There is a Sudoku board I would like you to look at

Click on this link

## ... by: Thomas

Then again, Hidden Candidates is also a covered subset, but instead of covering a subset of spaces with a set of possibilities, you cover a subset of possibilities with a set of spaces.

Sorry, I came here because I started to write a Sudoku solver and was wondering what inference rules it was missing (it's got hidden singles, naked * [because I enumerate all subsets], and pointing *). The first one I hit that it couldn't solve requires APE.

## ... by: Thomas

It's funny that you call these Naked Blanks, when they are covered subsets. Take your first example: the set of free spaces in row A is {A2, A3, A4, A5, A6}. The set of possibilities is {{1, 6}, {1, 6}, {1, 2, 5}, {1, 2, 5, 6, 7}, {2, 5, 6, 7}}. The subset {A2, A3} is covered by the set formed by the union of its composite sets of possibilities: {1, 6} U {1, 6} = {1, 6}. If any of the other free spaces (those in {A4, A5, A6}) were to be 1 or 6, we would have to come to the contradiction that either A2 or A3 has no possible candidate values, therefore A4, A5, and A6 cannot be 1 or 6.

I only point this out, because it is a lot easier to program when you're thinking in subsets.

## ... by: Edwin

I think you should illustrate a case here in "Naked Candidates" of a naked pair or triple where there are both green and black numbers appearing in the same cell!. In your examples above, there are cells with both yellow and black numbers, but no cell contains both green and black numbers. You need to show a case of green naked candidates when at least one appears in a cell along with other black numbers. FI, we could have two cells in a box or line with 12348 in one and 2379 in the other where no other 2's or 3's appear anywhere else in that box or line. Then one green 23 appears in one cell along with black 148 and the other green 23 appears in the other cell along with black 79. We can then remove any 2's and 3's in the other cells in that box or line. I recently ran into such a naked pair, and realized I had been looking only for naked candidates when no other numbers appeared in the same cells. IOW, 'naked' candidates don't have to be LITERALLY 'naked'! lol
Andrew Stuart writes:

Might you be think of Hidden pairs/triples/quads?

## ... by: JohnNoneDoe

The killer solver does not seem to recognize naked pairs linked by a cage.

There is a Killer Sudoku board I would like you to look at

Click on this link

There will be a naked pair in b6,c7 linked by the cage starting at b6. That pair can eliminate candidates in b7. The solver seems not to see this.

Andrew Stuart writes:

Thanks for this great example John. I have seen the gap in the search and plugged the omission. Good catch. Your example puzzle now completes. I have updated all solvers to 2.06.

## ... by: JohnF

I notice that the solver takes a noticeable amount of time (even on a fairly fast system) to perform the Naked/Hidden Quads test.

A quick way to speed up the solver would be to check whether there are at least 8 unfilled cells in the row/col/box under examination - if there are not, then (as you point out) the solver would already have found a Hidden/Naked Triple or Double.

In fact you can take this one step further; if you have checked for a Naked Quad and not found one, then you won't find a Hidden Quad unless all nine of the cells are still unfilled.

## ... by: David Spector

I believe that naked pairs also work if the conjugate pairs are located anywhere along an intersecting row and column. If two AB cells are located along an intersecting row and column, you can remove A and B from the intersection cell. This is because this cell will always "see" both A and B.

This situation is easy to find if you first find conjugate pairs, then see if they are on a row and column that intersect at a cell having two or more candidates.

Happens sometimes, not always.

## ... by: BabuYB

In the example for naked triple grid 1, the starting candidates for D8 are {5,6.9}; D9 => {1,5,6}; for E7 => {1,3,5,8,9}; E8 => {3,4,5,6,8,9} and E9 => {1,5,6,8}. However, this does not effect the explanation.

## ... by: Brian

In the second paragraph under Naked Pairs, you state: "It is clear that the solution will contain those values in those two cells." I don't think it is clear at all. By what logic can you rule out the possibility of 1 being in A4 or A5 at this stage of the solution?
Andrew Stuart writes:

If there are only two candidates left in two cells, they are compulsory. Unless you've marked up your candidates incorrectly on paper, these rule out other candidates in the same row/column/box.

## ... by: kasmar45

Is there any rule of thumb how to perform the first move on this scenario.

236 236 36

## ... by: Jan Bourdelle

Please tell me what the second paragraph under the second puzzle under Naked Triples means: thank you. Your site is wonderful

In terms of the candidates per cell, the column 1's triple is a {2/2/3} formation reading down and the second is {3/2/3}.
Andrew Stuart writes:

Merely that the cells contain 2 out of 3, 2 out of 3 and 3 out of 3 of the three numbers in the triple (in column 1) and similarly for column 9.

## ... by: jb681131

When the cells that form the Naked Subset are not only confined to one but to two houses (a row and a block or a column and a block), they are sometimes called a Locked Subset. Candidates can be eliminated from both houses.

## ... by: jeanne

Why is b1,2,3 not triple 568? I keep eliminating the wrong numbers

## ... by: marmal9174

Best display and information I have seen regarding Sudoku. Thank you for the clear pictures and explanation.

## ... by: Brian Fink

Regarding my previous comment, it doesn't matter how many cells are in the Naked Pair chain. There will be times when the pair itself will cancel out in the cell, because there is an odd number of overlapping Naked Pairs; but my example focuses on the quasi-complete Naked Pair vicious cycle, in which, if the cell in question was just that pair of digits, the puzzle would have 2 solutions. This trick eliminates that possibility from the equation.

## ... by: Brian Fink

I've discovered a version of Naked Pairs that is very helpful with just one cell (as opposed to an entire row/column/box).

Let's say we have a chain of an overlapping even number of Naked Pairs, all the same pair, with two ends that can see a single cell but are not both aligned on the same row or column with it or even sharing the same box. Then the pair can be removed from that cell and only that cell, leaving all other digits in that cell as possibilities.

For example, if we had C1=3/7/8, C7=7/8, E7=7/8, and E1=7/8, then in addition to removing 7/8 from Row C and Column 7, you may remove it from C1 as well.

There may also be a variation of this that only includes a cell that can be seen by two cells with naked pairs (same pair) that are not part of a traditional Naked Pair, but I have yet to prove that one.

## ... by: Jan

In the Naked Quad section is stated "Well, I can't find an example in my 2012 stock, ..."

But you do have one! In the Naked Triple example is also a naked quad for 2, 3, 8 and 9 in D9, E9, F9 and F8 (and so also a Hidden Quad for 1, 4, 5 and 7 in D7, D8, E7 and F7)

## ... by: Pete

Ahh, I get it now.....Square C1 contains only numbers which are part of the 4 numbers in the quad and no extra numbers. Whereas square B3 contains extra numbers which are not part of the quad.

It only took me a month to figure out.....

## ... by: Pete

In the naked quad example, I don't get how square c1 is identified as part of the quad, when square b3 contains 3 of the 4 quad numbers. The only thing I notice is that b3 doesn't contain number 1.
Is it the case that each of the quad squares has to contain one number which appears in all 4 squares? If so, then how does this tie-in with the naked triple possibility of (12) (23) (13) ? Here, there isn't a number which appears in all 3 cells..

Great site btw

## ... by: KeithD

Quick suggestion for people who still can't understand the naked triple explanations, above and in the comments: try using one of the three numbers in the triple as the solution for any of the candidates for elimination (in the row, column or box as appropriate). Now, only two of the cells in the triple can be solved, while the third has no valid candidate. Clearly, only the three triple cells can contain the three triple numbers.

Eg, in the first triple example, try putting 5 in E1. Now E5 is 8 and E6 is 9, but E4 has no candidate.

## ... by: Tim

Naked Pairs figure 2 shows (1,2) at A6 and G6.
Why not G4 and G6
Andrew Stuart writes:

That is a Naked Pair on G4/G6, but it doesn't lead to any eliminations (in the row) so it doesn't get highlighted.

## ... by: Klaus

I don't understand how the 1-2 pair in Fig.2 A-6 and G-6 can remove 1-2 in B5, can you please elaborate on this.
Thank You,
Klaus

BTY you have a great site, its so informative and I have learned so much from it, greatly appreciated.
Andrew Stuart writes:

Candidates 1/2 are the *only* candidates in cells A6 and G6. You notice that B6 has an extra candidate 8. If all three only had 1/2, we'd be in trouble - we'd have gone wrong on some previous step. 1/2 *must* go in A6 and G6, so it leaves no room for 1/2 elsewhere in the column. Fortunately, that 8 allows us to fill B6 and move on.

## ... by: Roy

Just realized there is another Naked Quad in Box 1 in the Naked Quad example (2,3,4,7) in cells A2, A3, B3 & C3! Alas, it doesn't really help reduce any other cells. With two Naked Quads in the Box, the only other number is 9 which has already been identified in cell C2.

## ... by: Roy

The logic of cell B2 being 1 or 8 for the Naked Quads section is derived from the fact one of the cells A1, B1 & C1 will contain either a 1 or 8 but not both as cell H1 will have the other. Cell B2 is the only other cell in this block that has either of these two numbers. I guess this would be a hidden pair within a pair of naked quads.

## ... by: Roy

Having reduced cell B2 to possibles 1 or 8, cell B1 can be reduced to possibles 5 or 6 (hidden pair with cell B4) and then H1 must be 8 as would be the case for B2! A1, B1 & C1 have been reduced to a naked triple (1,5,6) with a {2,2,2} formation!

## ... by: Roy

I believe you can go one step further with cell B2 in the Naked Quads example. It will have a value of either 1 or 8 given the restrictions from cell H1.

## ... by: Anon

i am really thankful to you for presenting this.really very useful content about sudoku solving. i am impressed.

## ... by: Peter Rogers

Hi
What a fabulous website. This is what I have been looking for for years a real how to solve sudokus. I am stunned at its teaching capacity.

## ... by: gerhard, sweden

Assume that a triplet consists of the three bigrams
(example) 56, 67, 57, occuring in one row.

Let´s say that the bigrams occur in region 1 (bigrams 56 and 67) and in another region (57).

It is obvious that they work the same as any true triplet, but less obvious that the figure 6 can be eliminated from the remaining squares of region 1.

If this is described somewhere else, please excuse me for commenting.

## ... by: Pieter, Newtown, Australia

Hi Andrew
I always love to double-check my solution to a puzzle using your solver. I got this one by XY-Chains but damn it! I missed the naked & hidden Quads, yet again! I usually do, damn quads! :-(

I noticed you don't have an example for quads in your "Pick an Example" drop-down list. Want to include this one?

It's from the Sydney Sun-Herald of 2011-09-4 (Auspac Media for the puzzle). You may need to check with them re copyright.

Thanks as always for your great solver!
Ciao, Pieter
Andrew Stuart writes:

Excellent example. Thank you for sharing. I don't think I can use it in my example list, but it can be linked here as you have stated the credits.

## ... by: Charlie R

Wow! What a site. I landed here by chance. I have been exploring sudoku myself, using my own excel-based solver, convinced that there must be a complete rule-based solution. I had found many of the rules myself, but this is a much more complete set, beautifully explained and illustrated. I bow to you, Oh Master

## ... by: Dayanandan

Landed on this site by chance. The joy experienced is such that I want to tell you this at once. This clears my doubt fully. I like this illustration as well.

Regards
Dayanandan

## ... by: hutch

i thought i could write this in excel and i did get some parts but soon realized the complexity and have stopped(at least 4 now). the stepwise debugger style is the bomb. i hope to improve my sudoku but i think i will spend a good bit of time just admiring this work.
many thanks for the obvious labor of love.
hutch
pawleys

## ... by: Andrew

Almost a year later, a response to Mike, who said:

"I notice that Sudoku Solver does not exhaustively identify all naked pairs as seen in the following puzzle.

http://www.sudokuwiki.org/sudoku.htm?bd=68050041905041000604160000000
9100040300700080400203960204871600000060104106000008

In row E the 2,5 pair in columns 3 and 7 should reduce cell E9 to just 1."

I don't understand why you say that. There's not such a naked pair there, and E9 has already a 3... Perhaps the sudoku saved with that id changed?

## ... by: Blaster88

Non seulement c'est génial mais en plus je bosse mon anglais !

## ... by: John

After reading this over, I think I understand why naked triples (and naked quadruples and quintuples). If you understand how naked pairs work, look at naked triples this way: When you solve one of the 3 cells, the other two cells become naked pairs or single. Then all three numbers in a naked triple can be eliminated from the other cells.

For example: (123) (123) (123)
Make any of the cells a 1: (123) (1) (123)
Drop 1 from the other cells: (23) (1) (23)
You can delete 1 from all other cells, because it is used. You can eliminate 23 from the other cells because it is a naked pair.

The same works for other triples: (123) (12) (23)
If the middle cell is 1: (23) (1) (23)
Eliminate 23 from other cells because it is a naked pair.
and so forth...

## ... by: Michael

To all who are having difficulty understanding this...

A naked pair shows the same two values and only those values in two different fields (in the same column, row, or three by three square). This shows that those two fields each must have one of the two values (there are no other values to choose from). Since a value cannot occur more than once in any one column, row, or three by three square) the two values can be safely removed from the other clues since it is know that they must appear in the place of the naked pair.

Naked Triples and Quads simply extend the same logic to 3 and 4 values.

## ... by: Mike

I notice that Sudoku Solver does not exhaustively identify all naked pairs as seen in the following puzzle.

http://www.sudokuwiki.org/sudoku.htm?bd=68050041905041000604160000000
9100040300700080400203960204871600000060104106000008

In row E the 2,5 pair in columns 3 and 7 should reduce cell E9 to just 1.
Andrew Stuart writes:

The solver is working correctly, but the behavior in these cases is worth explaining. As Naked Pairs are detected, the removal effects are applied. This might occasionally stop another Naked Pair from being found, since some numbers have been removed. The solver *could* detect all NPs and then apply the results simultaneously, but for speed and space I have chosen not to. Usually, the next set of NPs will be discovered in the next round. This applies to most of the basic strategies.

## ... by: Pete

I've been looking for help and this is the first I've seen that looks like it will help. Bring on the 6 star puzzles. I'm ready(I think).

## ... by: CS VIDYASAGAR

Excellent explanation with very useful examples to make one understand difficult concepts naked pairs and naked triples.
Thanks for keeping the aritcle simple and easily understandable.

## ... by: Harpo

I agree with buc; with the information given it still seems rather illogical to remove the other candidates.

## ... by: Werty

My explanation of naked triples.
On the example.
imagine that you put 5 in one of the columns 2, 3 or 4. That will leave only 7 and 8 as candidates in three columns - 1, 8 and 9. Clear?
You will get to similar wrong position when you put 8 in column 4.

## ... by: Carol Kennedy

I am just learning this game and so enjoy it. But I do not always understand your lessons. For example, if you have 4,8
4,8 in a row then you can eliminate the other 4,8s in that row, but can I also erase all the other 4,8s in the column and the entire box as well? Thank you.

## ... by: Curt Klemenz

I'm in same boat...having ultimate difficulty spotting hidden pairs and triples. When they are pointed out, .... I see them.

I suspect there is a mental algorithm for focusing attention toward the specific candidates, but no luck so far.

Anyone with a suggestion that's willing to share?

## ... by: Bruce D

An explanation on how the naked tripple works. As in the example shown, we have (7,8) (5,7,8) and (5,7,8). The first cell can contain a 7 or an 8. That means that the other two cells will then contain a 5 and 8 in the case the first one is a 7, or a 5 and 7 if the first cell is an 8. By having the last two cells being conditional on the other, we can eliminate the 5, 7, 8 from all other cells in the row.

## ... by: Rockmelon

I have been an accountant for 35 years (which means nothing) and I can't see the relationships among these numbers! I have a really difficuolt time understanding this and I love to do Sudoku!

Any suggestions??

## ... by: BobCarl

As you know, any row, column or box contains nine cells.

When there are only 3 different numbers that can fit into three of the nine cells, that automatically eliminates their use in the remaining six cells. Hence, they can be removed as candidates from those "other cells".

## ... by: buc

Re naked tripple: I would appreciate you explaining the logic of removing any of the three candidates from other cells.
Article created on 9-June-2005. Views: 878082