Naked triples and quads: the subset techniques

Naked triples and quads are the next step after pairs in the family of subset techniques. They look more complicated than they are. Once you can recognize them, puzzles that seemed stuck often start moving again. This guide explains the idea, shows it on diagrams and walks through a worked example.

What is a naked triple?

A naked triple is three empty cells in one row, column or box whose candidates together contain exactly three digits. Those three digits must fill the three cells between them, so no other cell in that unit can hold any of them. A naked quad is the same idea with four cells and four digits.

How it works

Cells are written as r#c#, row first and then column. So r4c6 is row 4, column 6. Rows are numbered from the top and columns from the left.

The method has three steps:

1. Find cells with few candidates. In a row, column or box, look for three empty cells that have only two or three candidates each.
2. Combine their candidates. Write down every candidate the chosen cells have between them. If the combined list has exactly three digits, you have a naked triple.
3. Remove those digits elsewhere. Remove the three digits from every other empty cell in the same unit.

The common misconception

Many players expect every cell in a triple to show all three candidates. That is not required. In the diagram below, the three cells hold {1,2}, {2,3} and {1,3}. Each cell has only two candidates, but together they use only the digits 1, 2 and 3, so they form a naked triple. Quads work the same way: four cells may each carry only two or three candidates, as long as their combined candidates are four digits.

The diagram shows a naked triple in row 1. The three cells marked x are r1c2 {1,2}, r1c5 {2,3} and r1c8 {1,3}. The digits 1, 2 and 3 are removed from the other cells shown. Cell r1c1 {1,4,7} loses the 1 and becomes {4,7}. Cell r1c4 {2,3,6} loses the 2 and the 3 and is left with the single 6. Cell r1c9 {3,5,9} loses the 3 and becomes {5,9}. Cells that do not change are left blank.

Hidden triples and quads

A hidden triple is the reverse situation. Three digits can go only in three cells of a unit, even though those cells also hold other candidates. Since the three digits must occupy those cells, every other candidate in them can be removed. Hidden quads work the same way with four digits and four cells.

The next diagram shows a hidden triple in row 1. The digits 1, 2 and 3 appear only in r1c2, r1c5 and r1c8. Their other candidates, 7, 9 and 6, can be removed. That turns r1c2 {1,2,7} into {1,2}, r1c5 {2,3,9} into {2,3} and r1c8 {1,3,6} into {1,3}. The result is a naked triple in the same cells.

How to spot it

• Start with cells that have two or three candidates. Most triples and quads are built from these cells. Scan a unit for several of them that sit together.
• Write the combined list before you trust it. Three cells with {1,2}, {2,3} and {1,2,3} make a triple on the digits 1, 2 and 3. If you count four digits across three cells, it is not a naked triple.
• Check which cells will change. Only cells outside the group lose candidates. The cells in the triple keep theirs.
• Look for a single after removal. A cell that drops to one candidate becomes a naked single, which you can place right away. In the first diagram, r1c4 becomes a single 6.
• Look in units with few empty cells. The fewer empty cells a unit has, the easier it is to test every combination. Quads are rarer than triples and are usually easiest to find in such units.

A worked example: a naked quad in a box

Take box 5, which covers rows 4 to 6 and columns 4 to 6. Suppose four of its empty cells have very few candidates:

• r4c4 {1,2}
• r5c5 {2,4}
• r6c4 {4,6}
• r6c6 {1,6}

Their combined candidates are {1,2,4,6}. That is four cells and four digits. So these four cells must hold exactly 1, 2, 4 and 6 between them, and no other cell in box 5 can hold any of those digits.

Now look at the other empty cells in box 5 that contain those digits:

• r4c6 {1,6,8} loses 1 and 6, so it is left with 8 and becomes a single.
• r5c4 {2,4,9} loses 2 and 4, so it is left with 9 and becomes a single.
• r6c5 {4,5,6,9} loses 4 and 6, so it becomes {5,9}.

The diagram shows the quad and the three cells it changes. Every x cell is part of the quad, and every - cell is in box 5 and loses the quad’s digits.

Once r4c6 and r5c4 are filled in, the rest of box 5 becomes much easier to finish. A quad often looks dull at first, but it removes a lot of candidates in a single step.

Common mistakes

• Assuming each cell needs all the digits. A naked triple is defined by the combined candidates, not by each cell carrying every digit.
• Adding a cell that does not belong. Every cell in the group must be in the same unit. A cell with a new digit breaks the group.
• Removing candidates from the group itself. The triple or quad cells keep their candidates. Only the other cells in the unit lose them.
• Mixing up naked and hidden patterns. A naked triple removes digits from other cells. A hidden triple removes other candidates from its own cells.
• Searching for quads too early. Check for pairs and triples in a unit first. Quads take more time to test and are rarer.

Practice naked triples and quads in Hush

Hush uses naked and hidden triples and naked quads at Hard. Hidden quads are so rare that no Hush puzzle needs one. The first 10 solved Hard or Expert puzzles are free, and Hush Plus, a one-time purchase at $4.99 with no subscription, unlocks unlimited Hard and Expert puzzles. Hush has no mistake limit, so you can test a triple, see what it removes and undo the move if you change your mind. Notes also clean themselves up when you place a number, which keeps the candidate lists easy to scan.

If you already know naked pairs and hidden pairs, triples will feel familiar. When you are comfortable with them, move on to the X-Wing, a pattern that works across two rows and two columns. To see where this level sits in the whole game, read about Expert sudoku or return to the technique guides.