Three digits with only three homes

The short version

A hidden triple is three digits confined to the same three cells in one unit. Every home for each of the three digits lies inside those cells, so those cells must hold those digits, so anything else pencilled into them is surplus.

The cells themselves may look busy — six or seven candidates between them. That is not the point. What is restricted is where the digits can go, not what the cells can hold.

The shape

Take a unit. Take three digits. For each of the three, mark every cell in the unit where that digit can still go.

If all three marks fall inside the same three cells, you have a hidden triple. The three cells belong to the three digits, and every other candidate in those cells can be erased.

The condition is strict: every home of every one of the three digits, inside those three cells. A single fourth home for any one digit breaks it.

Why it works

A unit contains each digit exactly once. Three digits need three cells. If all homes for all three digits lie inside a set of exactly three cells, those three cells are the only places the three digits can land — so they must take exactly one each.

Since each cell holds one value and there are three cells and three digits, the three digits fill the three cells between them. There is no room left for a fourth value. Every other candidate in those cells is therefore impossible.

Note the direction of the conclusion. You are not learning that a cell is 7 — you are learning that a cell cannot be 9. The placement comes later, often immediately, when one of the cells is left with a single candidate.

A real board

Row 4, three digits, three cells:

• In row 4, 3 can only go in R4C4 or R4C7.
• 6 can only go in R4C4, R4C7 or R4C8.
• 8 can only go in R4C7 or R4C8.
• All three digits' homes fall inside the same three cells: R4C4, R4C7, R4C8.

So those three cells belong to 3, 6 and 8. R4C7 loses its 9 — it was a candidate there and it is not any more.

This case is worth pausing on, because it is also a naked triple: the three cells' combined candidates are exactly {3, 6, 8}. The two descriptions are the same structure, read from opposite ends. That happens whenever the three cells have no candidates outside the three digits, and it is why the two techniques are so often confused.

How to scan for one

Hidden triples are found by digit, not by cell — the same sweep you use for hidden pairs, one level wider.

1. Take a unit. Rows are easiest to read, then columns, then boxes.
2. For each digit, list its homes. Not its candidates — its homes. Where in this unit can the digit still go?
3. Group by small home sets. Any digit with exactly two or three homes is a candidate for a pair or triple.
4. Test the union of cells. If three digits each have all their homes inside the same three cells, you have it. If the cells differ by even one, you do not.
5. Erase the surplus. Every candidate in those three cells outside the three digits goes.

The hard part is step 4, because you are comparing cell sets, not counts. Two digits confined to {A, B, C} and one confined to {A, B} is a valid triple. Two confined to {A, B, C} and one confined to {A, B, D} is not.

Hidden triple vs naked triple

The same duality as pairs, one level up:

A naked triple requires the cells to be poor: few candidates each. A hidden triple requires the digits to be poor: few homes each. The two conditions are independent, and both can hold at once — as in the board above.

If the hidden triple's three cells happen to hold nothing but the three digits, the naked triple fires too, and the eliminations overlap. That is not redundancy, it is the same fact stated twice.

Naked triples, the outward version.

The traps

Counting candidates instead of homes. The most common error. A cell reading {2, 3, 6, 8, 9} looks like it cannot be part of a triple, because five candidates is a crowd. But the question is whether 3, 6 and 8 have any home outside the three cells — the cell's own list is irrelevant.

One digit with four homes. Two digits perfectly confined, the third wandering. No triple. This is easy to miss because you usually find the two good digits first and stop looking.

Confusing a pair with a triple. Two digits confined to two cells is a hidden pair, and it only eliminates the other candidates from those two cells. Bringing a third digit in "for good measure" invalidates the whole thing.

Assuming elimination means placement. A hidden triple rarely places a digit on the spot. It is a setup move, and its payoff arrives one or two moves later as a naked single.

Where to use it

Hidden triples are rarer than hidden pairs and considerably harder to spot, which puts them in the band of techniques that appear when a hard puzzle has already resisted everything cheaper.

Their real value is diagnostic. If you have written full candidates, run the hidden-pair scan, come up empty, and the board is still stuck, running the same scan at width three costs very little and occasionally cracks the puzzle outright. It is the cheapest escalation left before you start counting fish.