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Before starting
1. The singles
1.1. Naked single
1.2. Hidden single
2. Locked candidates
2.1. Segment 1
2.2. Segment 2
3. The subsets
3.1. Naked subsets
3.1.1. Naked pair
3.1.2. Triplet et quad
3.2. Hidden subsets
4. Complex methods
4.1. Associated pairs (X-wing)
4.2. Multiple associated pairs (Swordfish)
4.3. Linked candidates(coloring)
4.4. Forced chains
5. Guesswork/Nishio

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Solving methods > 3- The subsets

3.1 Naked subsets

3.1.1 Naked pair
When a group contains two cells with the same pair of candidates (and only those two), then this candidates cannot be in another cell of the group. This is so for a row, a column or a block.

On the 2nd row there is the pair (1,4) in G2 and H2. One can deduce that if 1 is in G2, then 4 is in H2 or vice-versa. In any case these two cells will necessarily contain 1 and 4. One can thus exclude these candidates from the rest of the row.

As these pairs both belong to the 3rd block (upper right), then one can also eliminate 1 and 4 from the other cells in that block.


3.1.2 Triplet et quad
When three cells of one group do not contain other numbers than three candidates, those numbers can be excluded from the other cells of the group.
Notice:There is no need that these three cells contain all the numbers of the triplet! It's only necessary for these cells to be the only ones to have the three common numbers.

On this row, we have the triplet 1,4,6 in cells A,C and G or two candidates of this triplet. These three cells will necessarily contain the three numbers of the triplet. Therefore they cannot be elsewhere on the row and can be eliminated in the other cells (E and F).


Similarly for a quad, if four cells contain no candidates other than a determined quad, then these numbers can be excluded in the other cells of the group. As for the triplet, the cells containing the quad do not have to contain all four numbers of the quad.

More generally, for N numbers of candidates in a group, one has to find N cells that contain these candidates. Each cell must contain one of the candidates. For three cells containing a triplet, even an incomplete one, the three numbers of the triplet will be distributed in these three cells, and thus cannot be in the other cells of the group.

3.2 Hidden subsets

For the naked subsets (previous method: 3.1) the pairs, triplets and quads permit to exclude candidates from the other cells of the group.
With this method, the hidden subset, the subsets permit to exclude the other candidates from the cells wich contain them.

If there is N cells (2,3 or 4) containing N common numbers, then all other candidates for these cells can be excluded.

In this row, the pair (4,6) are only in the cells A and C.

The other candidates can thus be eliminated for these two cells, since they will necessarily contain either 4 or 6, and no other.


As for the naked triplet and quad (3.1.2)) the cells need not contain all the numbers of the triplet or quad. The hidden triplets are very difficult to spot. Fortunately, they are rarely usefull to solve a sudoku.
Hidden quads are practicaly impossible to spot !

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