Pairs and Multiples

 2 169 1356 13 3689 7 58 15 4 167 167 13567 4 368 16 258 125 9 14 149 8 15 2 159 6 3 7 47 8 79 6 5 3 1 49 2 16 3 169 7 4 2 59 8 56 5 46 2 9 1 8 34 7 36 9 5 4 8 7 16 23 26 13 8 2 167 135 369 4 579 569 15 3 167 167 2 69 1569 4579 4569 8

Pairs

There are two types of pairs, and this puzzle contains both types. A naked pair occurs whenever you have two cells in the same block (row, column or box) that can only hold the same two candidate values. Look at column 6. There are two cells, cell (2,6) and cell (7,6), that can only hold either a one or a six. If (2,6) is a one, then we know that (7,6) is a six, and vice versa. Because of this, we know that no other cell in column 6 can hold a one or a six.

A hidden pair occurs whenever you have two values in the same block that can only appear in the same two cells. Look at box 1 (in the upper left of the grid). The value three can only be located in cell (1,3) or cell (2,3). The value five can only be located in one of the same two cells. If you place the three in cell (1,3), then you must place the five in cell (2,3), and vice versa. Because of this, we know that cell (1,3) and cell (2,3) cannot hold any values except three or five.

 2 169 35 13 3689 7 58 (8) 15 4 167 167 35 4 368 16 258 125 9 14 149 8 15 2 59 6 3 7 47 8 79 6 5 3 1 49 2 16 3 169 7 4 2 59 8 56 5 46 2 9 1 8 34 7 36 9 5 4 8 7 16 23 26 13 8 2 167 135 369 4 579 569 15 3 167 167 2 69 59 4579 4569 8

Multiples

After making these candidate eliminations, let's look at row 1. The three blue cells can only hold the three values one, three and five. There are two possible ways to order these values within the three cells, but those three values will be placed in those three cells. Therefore, we know that those values cannot be placed in any other cell in row 1.

The same principle that we applied with naked pairs applies to naked triplets and naked quadruplets. It does apply to higher multiples, like quintuplets, but we don't usually use that terminology—at least not with classic sudoku. The reason is that naked multiples and hidden multiples are complementary. If there are six values to be placed in row 1, and three cells can only take three of the values, the the other three values must be restricted to the remaining three cells. That means that there is a hidden triplet in row 1 of this grid. If you look at the three yellow cells that haven't been determined, you'll see that the extra values are six, eight and nine, and those three values form a hidden triplet. Looking more closely, you can also see that the values six and nine form a hidden pair in cells (1,2) and (1,5). Whether you first notice the naked triplet or the hidden pair or the hidden triplet, you'll wind up at the same place. Cell (1,7) = 8, there is a 69 pair in cells (1,2) and (1,5), and the remaining three cells are a 135 triple.

You should understand that there is no particular pattern for the triple. The three cells may have candidate values 13, 35 and 15 or the may have 135, 135 and 135, or 135, 13 and 15, or any of the other combinations. It is only necessary that n cells be restricted to n values (naked multiple) or that n values be restricted to n cells (hidden multiple) within the same row column or box.

Sudoku Variants

This sudoku method works the same way for sudoku variants such as cube sudoku or the Sudoku of the Day puzzles. But there is an additional methods that is very much like naked multiples. Suppose that three cells are pair-wise neigbhors—even if they are not all in the same row, column or box. In a jigsaw sudoku, for example, cell A can be in the same box as cell B while cell B is in the same row as cell C, and cell A is in the same column as cell C. If those three cells are restricted to the same three values, then you can consider them to be an additional block, just like a row, column or box. If any value within this pseudo-block is restricted to part of the pseudo-block, then you can have block interactions between that subset and one or more other blocks.

Let's consider the case above with cells A, B and C. If A has candidate values 123, B has candidate values 12, and C has candidate values 13, then the values one, two and three will be distributed over the cells A, B and C. No two cells can hold the same value, because any two of the cells neighbors each other in a row, column or box. As a result, we can treat ABC as a new block, and see that either A or B will hold the two, while either A or C will hold the three. Therefore, we can eliminate the other occurences of two elsewhere in the box containing A and B. Similarly, we can eliminate the other occurences of three in the column containing A and C.

The same situation works for quadruples or more. The requirement is that you have a group of n cells that are pair-wise neighbors, and that the n cells are restricted to only n different values. That makes the n cells work like a block that can have block interactions with other blocks. (If you want to stretch it, you can have some of the cells not be neighbors, as long as the cells that are not neighbors don't have any candidate values in common. This can actually happen in cube sudoku, although I've never seen it.)