Hidden Pairs and Triples: Find Digits with Nowhere Else to Go

Read the digits' addresses

A cell with several notes can look unhelpful until you ask where those digits occur elsewhere. Hidden sets start with a house's missing digits and their possible locations. If two digits have only the same two cells available, those cells must accommodate them, even if extra candidates obscure the relationship.

HoDoKu defines hidden subsets through this restriction on locations. [1] The conclusion points inward: remove the unrelated candidates from the reserved cells. You are not yet assigning an order to the selected digits. The name describes where the restriction is easiest to see, not a special rule beyond ordinary Sudoku uniqueness within a house.

Uncover a pair in a busy row

Look for digits 2 and 7 in the first row model. Both occur only in c1 and c5. Column 1 lists {2,4,7}; column 5 lists {2,5,7}. The extra 4 and 5 are the distractions. Because the row needs both 2 and 7, the two cells must hold them in some order.

Remove 4 from c1 and 5 from c5. The lists become {2,7} and {2,7}. The other unresolved cells still carry combinations of 3, 4 and 5. You have separated the row into a two-digit reservation and the remaining work.

Conceptis illustrates the same inward cleanup for hidden pairs. [3] There is no additional elimination of 2 or 7 elsewhere in this row: their absence elsewhere was the premise that proved the pair.

A triple does not require identical position lists

For a hidden triple, collect the possible positions of three missing digits. Their combined set of positions must contain exactly three cells. The individual position lists may differ.

In our second model, digit 1 appears in c1 and c5; digit 2 appears in c1 and c3; digit 4 appears in c3 and c5. Their union is {c1,c3,c5}. No selected digit appears outside that group. These three cells therefore belong to digits 1, 2 and 4.

Remove the extra 5 from c1, 8 from c3 and 7 from c5. What remains is {1,2}, {2,4} and {1,4}. Stuart explicitly includes mixed-list triples of this form. [2] Insisting that every selected digit appear in all three cells would miss this example.

Build a small location table

Choose one row, column or box with reliable notes. For each missing digit, write down the cells where it remains possible. In the triple above, the table is short: 1 → c1,c5; 2 → c1,c3; 4 → c3,c5.

Now count distinct cells in the combined list, not repeated entries. Each chosen cell appears twice in that example, but there are still only three cells. Compare that count with the number of selected digits. Then inspect every other cell in the house to confirm that none of those digits has an extra home.

You can search just a few promising digits first. However, an unexamined cell cannot be treated as an excluded location. Hidden-set proofs require complete location information for the selected digits.

Do not clean the wrong side of the pattern

With a naked pair, two cells already show only two digits, so those digits are removed from the rest of the house. With a hidden pair, two digits already have only two locations, so other digits are removed from those locations. Saying the direction aloud helps prevent an accidental deletion.

An extra occurrence of a selected digit can invalidate the proposed hidden set. If 7 also appears in c9 in the pair example, 2 and 7 no longer have only c1 and c5 as their combined homes. You must reconsider the pattern rather than erase that inconvenient extra 7.

Do not choose the order inside the pair. After cleanup, {2,7} remains a pair; the hidden-set argument alone does not say which cell is 2.

Check what the deduction changed

These diagrams are row-level models with complete candidate lists. They show the smallest useful context for each proof, not a full puzzle or a promise that the same move is available on your current board. Enumerating their allowed row assignments confirms that the stated deletions discard no compatible arrangement.

After cleanup, inspect the affected columns and boxes as well as the original row. A smaller candidate list may help another technique, even when no immediate single appears. If nothing new follows, the elimination was still valid; do not strengthen its conclusion to manufacture progress.

Try the quiz, then compare this guide with naked pairs and triples. The two views ask different questions about the same shared-house constraint.