What is a Hidden Rectangle?
The hidden rectangle sudoku technique is a form of uniqueness-based elimination closely related to Unique Rectangle patterns. It appears when four cells form the corners of a rectangle (two rows × two columns) and a particular pair of digits appears as candidates in each corner, but one or more corners also hold extra candidates that “hide” the obvious two-digit pair. Under the standard Sudoku assumption that the puzzle has a single solution, that extra candidate can sometimes be eliminated because accepting it would create a non-unique solution.
Why this matters
Hidden rectangles are useful because they let you eliminate an extra candidate that otherwise looks harmless. This helps you avoid more complex chains or guesswork. The technique is safe only under the uniqueness assumption: if the puzzle is guaranteed to have one solution, some candidate choices would force two different valid completions, so those candidates cannot be true.
When to look for it
- Scan for pairs: When you notice the same two digits appearing together as candidates in multiple cells, check whether those cells sit on two rows and two columns that form a rectangle.
- Look for “almost-pairs”: If three cells have only the two digits and a fourth corner has those two digits plus one more candidate, that extra candidate may be eliminable.
- Early to mid solve: Hidden rectangles often show up after you have filled in many givens and reduced candidates, but before you start long chains or advanced techniques. If you are stuck and you see many 2-candidate cells, it’s a good moment to check for this pattern.
Basic logic (short)
Suppose digits A and B are the pair. The four rectangle corners are C1, C2, C3, C4. If three corners contain only {A,B} and the fourth contains {A,B,X} (with X ≠ A,B), then setting X in that fourth corner would leave the other three corners with A and B only—allowing two different placements of A and B inside that rectangle and producing more than one solution. Because we assume uniqueness, X cannot be true, so X can be eliminated from that corner.
A concise rule-of-thumb: three pure AB corners + one AB+X corner → eliminate X from the mixed corner.
Practical tips
- Confirm the “rectangle” geometry: the four cells must occupy exactly two rows and two columns (their intersections). They need not be in the same 3×3 box, but if two corners share a box the logic still applies as long as you can identify the four intersection cells.
- Check visibility: the cell that contains the extra candidate must see the other corners only in standard Sudoku houses (row, column, box). If eliminating X would create contradictions elsewhere, double-check before erasing candidates.
- Use it early: this is a simple and fast elimination. Before invoking longer chains, scan for candidate pairs in rectangle positions.
Example: step-by-step candidate-grid
Below is a small candidate-grid example that shows a hidden rectangle. The rectangle corners sit at (row 2, col 3), (row 2, col 7), (row 5, col 3), and (row 5, col 7). The pair to watch is 4 and 7. Three corners hold only {4,7}; one corner holds {4,7,9}. The extra 9 hides the pure rectangle. Under the uniqueness assumption, 9 can be eliminated from that corner.
How to read the example
1) Identify the four intersection cells and spot the common pair (4 and 7).
2) See that three corners are pure {4,7}. The fourth corner (row 2, col 7) has {4,7,9}.
3) Ask: if that corner were 9, would the rectangle be forced into two interchangeable 4/7 placements? Yes — the remaining three corners would still allow two different assignments of 4 and 7 in the rectangle, creating multiple solutions.
4) Because the puzzle must be unique, 9 cannot be true in that corner. Remove 9 from (row 2, col 7). That often unlocks a straightforward next step: a single remaining candidate somewhere or a new elimination.
Caveats and common mistakes
- Don’t eliminate a digit unless the rectangle condition (three pure corners + one mixed corner) is strictly met. If two corners have extra candidates, Type 1 logic does not apply.
- Hidden rectangles rely on uniqueness. If you’re solving a puzzle that might have multiple solutions (many online generation modes or some intentionally ambiguous puzzles), the elimination is not logically forced.
- Double-check peer interactions: removing the extra candidate might not immediately yield a number, but it can create chains or singles elsewhere — be ready to recompute candidates.
Wrapping up
The hidden rectangle sudoku technique is a compact and powerful removal trick. Keep an eye out whenever you see the same two digits appearing across two rows and two columns. If three corners are pure and one corner is mixed, you can safely eliminate the extra candidate in that mixed corner under the usual uniqueness assumption. It’s a small pattern that often produces outsized progress when you’re stuck.