Sudoku Algorithms

Naked Triples are very similar to Naked Pairs. As the name indicates, there is a list of three candidates, and are three cells in the same block, row, or column, which have no other candidates in these three cells except those from the list. Unlike Naked Pairs, it is not required for Naked Triples that all three sells have all tree candidates, but each of selected candidates should be presented in at least one of these cells. A picture on the left represents an example of Naked Triples in square (2x1). There are three candidates '4', '7', and '9', marked yellow, in the cells D1, D2, and D3. We cannot tell which number will occupy any of these three cells, but since these numbers cannot appear in any other cell of the same block, we can remove hints '4' and '7', marked red, from the cell E1 and a hint '9', marked red, from the cell F1.

The picture on the left shows an example of the Naked Triples in a row. Since cells E4, E5, and E6 are in the same row E, and they all share the same list of three candidates '2', '7', and '9', marked yellow, we can remove hints '2', '7', and '9', marked red, from cells E2, E3, E8, and E9 in the row E.

The next picture on the left shows an example of the Naked Triples in a column. Since cells B3, D3, and G3 are in the same column 3 and all share the same list of three candidates '2', '7', and '9', marked yellow, we can remove hint '2', marked red, from cell G2.

A picture on the left shows an example of a Naked Quads in a row. Naked Quads are similar to Naked Triples. The only difference is that there is a list of four (not three) candidates in Naked Quads, and there are four cells in the same block, row, or column with no other candidates in these four cells except those from the list. As in the case of Naked Triples, each selected candidate in the Naked Quads should be presented in at least one of the four cells. In the picture above, the Naked Quad is in the row H. Four candidates, '3', '4', '6', and '9', marked yellow, are in the cells H6, H7, H8, and H9. These numbers cannot appear in any other cell of the same row, so we can remove hints '3', '4', '6', and '9', marked red, from the cells H1, H3, and H5.

The picture on the left illustrates the Naked Quad in a square example. The four cells (D4, E4, E6, and F4) in the central square have possible candidates from the following list: '1', '3', '6', and '9' (marked in yellow). These candidates should not appear in any other cell of the central square. We can remove candidate '1' (marked in red) from cells D5, F5, and F6, and candidate '3' (marked in red) from cell F6.

The picture on the left shows an example of the Naked Quad in a column. Four candidates, '1', '3', '8', and '9', marked yellow in four cells, B2, E2, F2, and I2, present a Naked Quad in the second column. These candidates should not appear in any other cell in the second column. Candidates '1' and '8', marked red in cell C2, can be removed.