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Naked Candidates
'Naked' in this context refers to all the remaining possble 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 futher 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 columns, 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 but you reason "well, X can't go anywhere else in either the row, column or box, so it must go here.
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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, box or column.
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.
Figure 1
Consider this center box in Figure 1. There are two 4/7s at A and B. Two other cells contain 4s and 7s. We remove those to produce the right hand picture.
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Figure 2 to the right is the same example but we're looking down the column at our two 4/7s at A and B. In the box below are two cells C and D which also contain 4s and 7s. We can safely remove the 4 from C and the 4 and 7 from D.
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Figure 2 |
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.
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The combinations of candidates for a Naked Triple will be one of the following:
The last case is interesting and the advanced strategy XY-Wings uses this formation.
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(123) (123) (123)
(123) (123) (12)
(123) (12) (23)
(12) (23) (13)
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To see a Naked Triple in action look at this center strip from an example board:
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We have a triple in columns 1, 8 and 9. There are three other squares with 5,7 and 8 so we can
clear them off leaving:
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Naked Quads
A Naked Quad is rarer, especially in its full form but still useful if they can be spotted. The same logic from Naked Triples applies.
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We have a Naked Quad arranged nicely together in the top row. Because we have 2/4/7/8 in columns 3, 4, 5 and 6 (marked in green circles) we can clear all other occurrences from the row (marked in red squares).
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Naked Quad Example: Load Example or : From the Start |
Comments...
... 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 being found since some numbers have be 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: maurice ackroyd
Please modify your text so that the triple 5,7,8 is shown both before AND after 'removal'. The situation at present (12 May2009) is confusing. Thanks. Also, can you point to a strategy for 'manua'l sudoku solving. By which I mean without unnecessary entering of all possible candidates. - all very well if you have a computer solver and don't mind entering a puzzle in its entirety. Thanks again.
... 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.
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