Hidden Triples in Sudoku: How to Spot Them and Why They Are Rare

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Published April 14, 2026 · Updated October 14, 2026

In box 1 of Evil puzzle 458, drawn after 18 moves in the first pair of grids, eight cells are empty. Ask where the 4 can go: R2C1, R2C3 or R3C3. Ask the same of the 6: the same three cells. The 9 has only two of them, R2C3 and R3C3. Three digits with three cells between them use those cells up, so whatever else is pencilled into them can go: the 1 and the 8 in R2C1, and the 3s in R2C3 and R3C3. That is a hidden triple.

Puzzle 458 is one our Evil sudoku page can deal you, and you will rarely need a hidden triple even there. Our Easy, Medium, Hard and Expert levels never do, and only 160 of our 5,000 Evil puzzles do, fewer than need a naked pair, a hidden pair or a naked triple. That is not because the pattern is scarce. Most hidden triples are another move in disguise.

What is a hidden triple in sudoku?

A hidden triple is a set of three digits with only three possible cells between them in one row, column or box (a house). Each of those cells must end up as one of the three digits, so every other candidate in the three cells can be crossed out. Nothing is removed anywhere else in the house. Hidden pairs, triples and quads go by one name, hidden subsets.

“Hidden” is about how it looks. R2C1 shows 1, 4, 6 and 8, and R2C3 shows 3, 4, 6 and 9, so no cell gives the pattern away. You find it by asking where each digit can go, not what each cell holds. Three cells that hold only three digits between them are the other pattern, a naked triple.

Why is it safe? Suppose R2C1 were a 1. The 4, the 6 and the 9 would then all need R2C3 or R3C3: three digits, two cells. A 3 in R2C3 or R3C3 fails the same way. Whichever cell gets which digit, the three cells hold 4, 6 and 9 and nothing else.

To watch our helper find it, open puzzle 458 in the Sudoku Helper; one press of Next Hint gives the pencil marks, then alternate Next Hint and Apply Hint until the nineteenth hint, this triple (“Step 19: Hidden Triple found. (6 techniques available)”). On the way, two pointing moves from box 4 clear the 1s and 8s from the top of column 3. Pencil the board in from the digits alone and those marks are still there, but the triple holds all the same.

How do you spot a hidden triple?

Work one house at a time and list, for each digit it still needs, the empty cells that can take it. Box 1 of puzzle 458, at the first figure:

Evil puzzle 458 after 18 moves. Box 1 has only its 7 so far, so it needs the other eight digits. The highlighted rows are the triple, the only one in the box.

Six digits have two or three places. The 1, the 2 and the 9 have two each, but no two of them share the same two cells, so there is no hidden pair to take first. Start from the 9: its cells are R2C3 and R3C3, and the 4 and the 6 add just one more, R2C1. Start from the 1 or the 2 instead and any digit you add takes you past three cells. As a routine:

1. Pick a house with seven or more empty cells, two digits placed at most. In a house with five or six, a hidden triple is always a naked pair or naked triple seen from the other side (see below), so once you have checked for those there is nothing new to find. Every one of the 165 hidden triples our helper used across the 5,000 Evil puzzles sat in a house with seven or more empty cells.
2. Count each missing digit’s places and set aside any digit with four or more.
3. Start from a digit with two places and look for two more digits whose places add at most one new cell. Of those 165 triples, 158 had a two-place digit to start from.
4. Cross every other candidate out of the three cells. Leave the rest of the house alone.

Scan the boxes as well as the lines. Of the 165, 120 lay in a row or column, but 45 were in a box alone, their cells not in one row or column, like box 1’s.

Do the three digits have to be in all three cells?

No, and usually they aren’t. In box 1, R2C1 has the 4 and the 6 but no 9. Row 8 of Evil puzzle 109 goes further: none of its three cells holds all three digits.

Each of the 1, the 6 and the 9 has exactly two places in row 8, and no two of them have the same two, so there is no hidden pair. Follow them round, though, and they close up: the 6 joins R8C1 to R8C9, the 1 joins R8C9 to R8C5, and the 9 joins R8C5 back to R8C1. Three digits, three cells.

That shape is common. Of the 165 hidden triples our helper played, 47 looked like row 8, every cell holding two of the digits and every digit two places. Only 7 had all three digits in all three cells; the other 111 sat in between, 64 with one full cell and 47 with two.

Hidden triple or naked triple: what’s the difference?

They push in opposite directions. With a naked triple you find three cells whose candidates come to just three digits, and you strike those digits everywhere else in the house. With a hidden triple you find three digits that fit just three cells, and you strike everything else out of those cells: the eliminations land inside the pattern, not around it.

They are also one deduction read from opposite ends, as our guides to hidden pairs and naked triples show. In a house with n empty cells, a hidden triple leaves the other n − 3 cells holding nothing but the other n − 3 digits, a naked set that strikes exactly the same candidates. In box 1 of puzzle 458 that is R1C1, R1C2, R1C3, R2C2 and R3C2, which hold only 1, 2, 3, 5 and 8: five cells to check instead of three. In row 8 of puzzle 109 it is a naked quad, R8C3, R8C6, R8C7 and R8C8 holding only 2, 3, 5 and 7. In a house this empty the hidden reading is the smaller one, and the one to look for.

Why are hidden triples so rare?

Our helper looks for hidden triples last of the four subset moves, after naked pairs, hidden pairs and naked triples, and that other reading decides how often anything is left for it. Where a house has five empty cells, a hidden triple is a naked pair read the other way; where it has six, a naked triple. Both are checked first. (With four, the other side is a naked single.) Only from seven empty cells up is the other side a naked set of four or more, which the helper never checks.

Every hidden triple that would remove a candidate, at the 4,618 positions in the first 500 Evil puzzles where no single was left (1,074 had at least one), walking simplest first with the 23 techniques we measure. Our helper checks naked pairs, hidden pairs and naked triples before hidden triples, and no larger naked set. Measured October 2026.

So 1,242 of the 1,586 (78%) were a naked pair or naked triple in disguise. Only 291 positions had a hidden triple in a house of seven or more empty cells, and at 269 of those something simpler was available too. A hidden triple was the move at 22 positions, on 21 of the 500 puzzles. Across all 5,000 the helper played 165, every one in a house with seven or more empty cells (134 with seven, 29 with eight, 2 with nine).

None of this reaches our classic levels. Across 800 of their puzzles, 200 per level, played simplest first, not one needed a hidden triple. The generator sees to that: it deals a puzzle only when the techniques allowed at that level solve it, which means singles up to Medium, locked candidates from Hard and naked pairs from Expert. Our Expert puzzles had a hidden triple somewhere at 226 of their 277 positions without a single, yet there was always a simpler move as well. Guides that file hidden triples under medium or hard puzzles are not describing ours.

What does a hidden triple unlock?

The payoff, when there is one, comes from the struck digits, in houses the triple never mentions. In puzzle 458 the 1 that left R2C1 had one other place in row 2, R2C4, and one in column 1, R1C1, which makes both of them hidden singles; they are blue in the second figure. A quick payoff like that came after 43 of the 165, every time as a hidden single of a struck digit, never as a naked single. The typical triple removed four candidates, the biggest eight.

Don’t look for more inside the triple’s own house. Its three cells are now a naked triple, but no other cell there holds their digits, and solving simplest first it removed nothing after any of the 165. The struck marks matter later, too: puzzle 458 still needs an XY-Wing and a W-Wing, and the W-Wing stands on R2C1, the cell the triple cut down to 4 and 6. After the triple the helper needs 48 more moves to finish, 45 of them singles.

Out of pairs and naked triples? Where to look next

Give hidden triples a quick look in houses with seven or more empty cells, then move on. We found 2,043 positions in the first 500 Evil puzzles with singles, locked candidates, both kinds of pair and naked triples all used up. A hidden triple was there at 22 of them, about one in a hundred.

Nor will one usually finish the job. Of the 160 puzzles that need a hidden triple, it is the hardest step on only 7; the other 153 need something harder or beat the helper altogether. After a puzzle’s first hidden triple, the next harder step was most often an almost locked set (43 puzzles), a W-Wing (28), an XY-Wing (28) or a Skyscraper (13). When the helper’s eighteen run out, as on 46 of the 160, chains come next: X-cycles, XY-chains, the BUG check and forcing chains. The order we suggest for all of them is in our guide to sudoku techniques.

Four digits shut into four cells make a hidden quad, which works the same way. Our helper doesn’t look for them, and the hidden pairs guide explains why little is lost.

How can you practise spotting hidden triples?

Practise on Evil puzzles, and fill in every pencil mark first; the Start Notes button does it in one press. The move rests on digits being absent from the rest of the house, so one mark you forgot to write can invent a triple. Each time the singles dry up, pick out the houses with two digits or fewer placed and count places. The Sudoku Helper’s Next Hint offers a hidden triple only when nothing simpler is on the board, so if it shows you one, you haven’t missed an easier move.