Naked Pairs, Naked Triples, Naked Quads Strategy

In sudoku, you will sometimes reach a point where you can't solve a square, but you can narrow down the possible numbers in that square, sometimes referred to as candidates. In some cases, you can make logical conclusions from those candidates that will help you solve the rest of the puzzle.

Naked Pair

In sudoku, a naked pair is a set of exactly two candidates that are in exactly two squares in a row, column, or block. For example:

In this example, the 2 and 3 in red on the right side is a naked pair. One of the squares in the pair has to be 2, and one of the squares in the pair has to be 3. If a square that is not part of the naked pair were set to 2, for example, then both of the squares in the naked pair would have 3 as the only candidate, which would be invalid.

Because of this, you can remove 2 and 3 as candidates from the other cells in the row. The 2 and 3 in the other cells have been crossed out in red in this example.

Interactive Example

The following interactive puzzle has several naked pairs that can be used to eliminate possible candidates. Click the button to see an explanation of the answers:

The 3 and 6 in the second row are a naked pair, so you can remove 3 and 6 from the 3, 6, 8 in that row, leaving only 8 and solving that square.

The 1 and 3 in the third row are a naked pair, so you can remove 1 from the 1, 8 in that row, leaving only 8 and solving that square.

The 1 and 4 in the fourth column are a naked pair, so you can remove 1 and 4 from the 1, 4, 8 in that column, leaving only 8 and solving that square.

The 1 and 5 in the ninth column are a naked pair, so you can remove 1 from the 1, 3 in that column, leaving only 3 and solving that square.

Puzzle string:

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Naked Triple

Naked triples are harder to spot. Sometimes, all three numbers in a naked triple appear in all three squares of the triple. However, sometimes the numbers of the triple don't all appear in all three squares, as in the example below.

The squares that have 7, 8, and 9 in red form a naked triple. Notice that if a square other than one of the naked triple squares were set to 7, for example, then the three naked triple squares would only have 8 and 9 as their candidates. The square that has 7, 8, and 9 in it would have both 8 and 9 as its remaining candidates, but each of the other two naked triple squares would be set to 8 and 9, respectively. This would be invalid, because the square with 8 and 9 as its candidates would no longer have any valid options.

Because of this, you can get rid of 7, 8, and 9 from the other cells in this row.

Interactive Example

The following interactive puzzle has several naked triples that can be used to eliminate possible candidates. Click the button to see an explanation of the answers:

The third row has a naked triple consisting of 1, 4, and 5. Because of that, you can remove the 4 from the 4, 9 in the second column of the third row, and the 1, 4, 5 from the ninth column of the third row.

The sixth row has a naked triple consisting of 1, 2, and 8. Because of that, you can remove the 2 from the 2, 9 in the first column of the sixth row.

The eighth row has a naked triple consisting of 3, 5, and 7. Because of that, you can remove

• the 3 and 5 from the first column of the eighth row
• the 3 and 7 from the third column of the eighth row
• the 3 and 5 from the sixth column of the eighth row
• and the 5 from the ninth column of the eighth row

The first and second columns both have a naked triple consisting of 3, 4, and 5. Because of that, you can remove the 3 and 5 from the eighth and ninth rows of the first column, and the 3, 4, and 5 from the ninth row of the second column.

The sixth column has a naked triple consisting of 4, 5, and 8. Because of that, you can remove the 5 from the fifth row of the sixth column, and the 4 and 5 from the eighth and ninth rows of the sixth column.

The upper right box contains a naked triple consisting of 4, 5, and 8. Because of that, you can remove the 4 and 8 from the middle square on the right side of the box, and you can remove the 4 and 5 from the bottom right square of the box.

Puzzle string:

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Naked quads are rare, but they can occur. They can be difficult to find, however, unless you specifically look for them.

In the example above, the numbers in red form a naked quad. The four squares that contain the red numbers only contain numbers 2, 4, 8, and 9. That means that those four squares MUST contain those four numbers. Because of that, you can eliminate the 4 as shown crossed out in red.

Interactive Example

The following interactive puzzle has several naked quads that can be used to eliminate possible candidates. Click the button to see an explanation of the answers:

The ninth row has a naked quad consisting of 2, 4, 6, and 8. Because of that, you can remove 4 and 8 from the third column of the ninth row, solving that square.

The sixth column has a naked quad consisting of 2, 6, 8, and 9. Because of that, you can remove 2 and 8 from the fifth row of the sixth column, solving that square.

Puzzle string:

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Example Puzzle

Try solving the following interactive puzzle. It can be solved using only the Naked Pairs, Naked Triples, and Naked Quads strategies.

Puzzle string:

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