Sudoku technique

Naked Quad

Four cells in the same zone share only four possible digits between them. Those four digits are reserved for them. Rare in practice: there are a lot of combinations to explore, and the engine only reaches for it as a last resort before the fish techniques.

A puzzle that requires it

Level Expert — the puzzle is shown twice: right below, and again in the demo that follows.

						1	7
				6	9		8
1					7	2
	3		7					5
	2	1		9		8	6
5					6		4
		4	6					1
	1		4	3
	5	7

Demonstration

State of the grid right when the technique applies, after 10 earlier deductions.

Demo

Demo — watch first, you'll play right after.

Technique's pattern
Candidates removed
Solved earlier
Given (clue)

			2			1	7
			1	6	9		8
1					7	2
	3		7			9	1	5
	2	1	5	9		8	6
5				1	6		4	2
		4	6	7				1
	1		4	3		5		9
	5	7	9		1	3		3

Quadruplet nu

Ces 4 cases n'acceptent que 2, 3, 5, 9. À elles quatre elles épuisent ces chiffres : personne d'autre dans la zone ne peut les prendre.

Candidates removed:

the 5 from cell L8C7
the 9 from cell L8C9
the 3 from cell L9C7
the 3 from cell L9C9

Your turn: puzzle 1 →

The Naked Quad pushes the logic of the Naked Pair and the Naked Triple to four cells. Four cells of the same zone whose candidates, taken together, make up only four digits share those four digits between them. You can erase them from every other cell of the zone: the Naked Quad eliminates elsewhere.

The principle

A zone is a row, a column or a box. Take four of its cells and combine their candidates. If that combination holds only four digits, say 1, 4, 6 and 9, then those four cells share those four digits, one each. You do not know who takes what, but between them, they use them up.

In the other cells of the zone, those four digits are therefore impossible. Remove them from their notes. The cells of the quad do not change: the cleanup happens in the neighboring cells.

As with the triple, each cell does not need to contain all four digits. One cell may have only 1 and 4, another 4 and 6, a third 6 and 9, a fourth 1 and 9. What counts is the total: four cells, and four digits in all, not one more.

How to spot it

You need candidate notes. In a zone, look for cells with few candidates, two, three or four. Pick four and check that the combination of their notes fits in four digits. If a fifth digit shows up, there is no Naked Quad. Otherwise, erase the four digits from the other cells of the zone.

This is the most tedious of the naked groups: there are many sets of four cells to compare. The complement shortcut helps a lot. In a zone with n empty cells, a naked group of k cells always matches a hidden group of n − k cells. With six empty cells, a Naked Quad equals a Hidden Pair on the other two cells. With seven, a Hidden Triple.

A good habit: if you already see a Hidden Pair or Triple, there is no point hunting for the matching Naked Quad — you already have the equivalent elimination.

A worked example

On row 5, three digits are already placed: 3, 5 and 8 (in R5C2, R5C5 and R5C8). The six empty cells therefore need to hold 1, 2, 4, 6, 7 and 9.

Four of those cells have tight candidates: R5C1 holds 1 and 4, R5C3 holds 4 and 6, R5C6 holds 6 and 9, R5C9 holds 1 and 9. Their combination is exactly {1, 4, 6, 9} — four digits, four cells: a Naked Quad, though no cell contains all four digits.

The other two empty cells of the row are R5C4 with candidates 1, 2, 6 and 7, and R5C7 with 2, 4, 7 and 9. Remove 1 and 6 from the first, 4 and 9 from the second: 2 and 7 remain in each. A Naked Pair 2-7 shows up on those two cells — it is also the Hidden Pair that complements the quad.





3	5	8

Naked Quad {1,4,6,9} on row 5: the four yellow cells hold only those four digits between them; on the two red cells, {1,6} and {4,9} are eliminated, leaving the pair 2-7.

Eliminations and placements

In the example, the Naked Quad removes 1 and 6 from one cell, 4 and 9 from another. Both become {2, 7}: a new Naked Pair. The quad placed no digit, but it created a pair that, in turn, is going to clean 2 and 7 elsewhere in the zone and in the two shared zones.

That is the signature of the quad: it never lands on a direct placement, but it dramatically simplifies the zone, and a simpler deduction almost always fires on the next turn.

Common mistakes

The most common mistake is to believe that every cell must contain all four candidates. That case exists, but it is rare. It is enough that the combination of the candidates of the four cells makes exactly four digits.

Another trap is to count a cell that brings in a fifth digit. One cell too many and the combination goes past four digits: the quad no longer exists. You also really need four cells: if three cells are enough for three digits, it is a Naked Triple, simpler to handle.

Finally, do not mix it up with the Hidden Quad, which is recognized by its digits and cleans its own cells. The Naked Quad is recognized by its cells and cleans the neighbors.

Related patterns

The Naked Triple applies the same reasoning with three cells and three digits. It is much more frequent than the quad, and often easier to see: fewer combinations to try.

The Hidden Pair is the inside view of the Naked Quad when the zone has six empty cells. The two remaining digits, outside the quad, can only fall in the two cells outside the quad: spotting them there is the same elimination, reached from the other side.

The Hidden Quad is the dual pattern: four digits can only land in four cells, which then monopolize those digits. The other candidates of the four cells disappear.

When to use it

The Naked Quad is expected at the Expert level, along with both triples, the Hidden Quad and the X-Wing. In practice, it is rare: the Solvissima engine walks through techniques in order of increasing weight, and a simpler deduction (single, pair, triple) usually unlocks the zone before it needs to look at a quad.

If you want to meet it in practice, load an Expert puzzle where no other technique unlocks — the matching puzzles are flagged in the Learn catalogue.

Frequently asked questions

Must every cell of a Naked Quad contain all four candidates?

No. It is enough that the combination of the candidates of the four cells makes exactly four digits. A cell can have only two, as long as the four cells together do not go past four digits in total.

Why is the Naked Quad rare?

There are many groups of four cells to examine in a zone, and simpler techniques usually unlock the grid first. In a zone with few empty cells, it often coincides with a Hidden Pair or Triple, which is easier to see.

How are naked groups and hidden groups related?

In a zone with n empty cells, a naked group of k cells implies a hidden group of n − k cells, with the same eliminations. For example, with six empty cells, a Naked Quad equals a Hidden Pair on the other two cells.

Does a Naked Quad place a digit?

No, it removes candidates from the other cells of the zone. Those eliminations can then leave a Naked or Hidden Single, which lets you place a digit.

How do you tell a Naked Quad from a Hidden Quad?

The Naked Quad looks at cells: four cells whose combined candidates make exactly four digits. It eliminates those digits elsewhere. The Hidden Quad looks at digits: four digits that can only land in four cells. It eliminates every other candidate inside those four cells.