Hidden Pairs: Revealing Numbers That Hide Together

Hidden Pairs build on the logic of Hidden Singles but involve two numbers instead of one. The discovery: when two specific numbers appear as candidates in only two cells within a row, column, or 3×3 box — and those two cells are the same two cells for both numbers — you've found a Hidden Pair. Those two numbers are "hidden" because the cells may also list many other candidates, but once identified, you can safely eliminate all other candidates from those two cells, leaving only the pair.

How Hidden Pairs Work

The process is slightly more involved than Hidden Singles, but the payoff is often bigger:

• Step 1: Choose a row, column, or 3×3 box to analyze.
• Step 2: For each number 1-9, list every cell in that unit that contains the number as a candidate.
• Step 3: Look for two different numbers that each appear in exactly two cells — and the two cells are the same for both numbers.
• Step 4: These two cells form a Hidden Pair. The two numbers must occupy these two cells. Eliminate all other candidates from those two cells.

After elimination, the two cells will now each show only the two numbers of the Hidden Pair. They've become an Obvious Pair — allowing you to then eliminate those two numbers from every other cell in the unit, just as with Naked Pairs.

A Concrete Example

Let's work through a realistic scenario in a 3×3 box. The candidate distribution for each number in the box looks like this:

• Number 1 appears as a candidate in cells: A, B, D, F, H
• Number 2 appears as a candidate in cells: A, C, F
• Number 3 appears as a candidate in cells: C, F
• Number 4 appears as a candidate in cells: C, G
• Number 5 appears as a candidate in cells: B, C, E
• Number 6 appears as a candidate in cells: A, D
• Number 7 appears as a candidate in cells: A, D
• Number 8 appears as a candidate in cells: H, G
• Number 9 appears as a candidate in cells: F, H

Look closely at numbers 6 and 7. Both appear in exactly two cells: A and D. These are the same two cells for both numbers. Cells A and D form a Hidden Pair for 7. This means cells A and D must, between them, hold the 6 and 7 for this box. No other candidate matters in these two cells — we can eliminate everything except 6 and 7 from A and D.

Before elimination, cell A might have had candidates 9 and cell D might have had 8. After eliminating the extra candidates, cell A becomes 7 and cell D becomes 7. We've transformed a cluttered pair of cells into a clean Naked Pair. From here, we can eliminate 6 and 7 from all other cells in the box — cells B, C, E, F, G, and H lose 6 and/or 7 as candidates.

Hidden Pairs vs. Obvious Pairs

The relationship between Hidden Pairs and Obvious Pairs is beautifully symmetric:

• Obvious Pairs: The two cells show only the pair numbers (naked). You eliminate the pair numbers from other cells in the unit.
• Hidden Pairs: The two numbers each appear only in the same two cells (hidden). You eliminate other candidates from those two cells.

After applying Hidden Pairs, the two cells become an Obvious Pair. After applying Obvious Pairs, you may reveal Hidden Singles or Hidden Pairs in neighboring cells. The techniques are designed to work together in a continuous cycle of elimination and revelation.

When to Use Hidden Pairs

Hidden Pairs are an advanced technique that becomes essential on Expert and Extreme puzzles. They are harder to spot than Obvious Pairs because you must scan the entire unit for each number's candidate pattern, rather than just looking at individual cells. However, the reward is substantial: a single Hidden Pair often clears 5-10 extraneous candidates from the board, triggering a cascade of Obvious Singles and hidden Singles.

A systematic approach helps. When you're stuck, scan each unit with this question: "For each number 1-9, which cells could possibly hold it?" Write down (or mentally track) the list. Look for two numbers that share exactly the same short list of cells. That overlap is your Hidden Pair. For maximum efficiency, start with rows and columns that have many empty cells but relatively few candidate distributions — densely populated units are less likely to contain Hidden Pairs.