What the empty rectangle is

Starting grid for the example (zeros are empty cells). Few givens are shown to keep the focus on the candidate pattern.
Candidate map for digit 7. Four corner candidates form the empty rectangle across the top-left and top-middle boxes; the right-top box has no 7 in row 2 (box-line absence). A 7 candidate in the center band is shown as eliminable because it sees both rectangle corners.

Empty rectangle is a pattern for eliminating a candidate for a single digit. It combines a box-line absence (a digit cannot appear in a particular row or column inside one 3x3 box) with candidates that are confined to two lines in the two adjacent boxes. When those two pieces line up, you can remove the digit from a cell that sees both ends of the resulting strong link.

Why this matters

Empty rectangle is a neat one-digit trick: it doesn't require advanced chains or large nets, just attention to candidate shapes inside a band or stack of three boxes. It is especially useful in medium-to-hard puzzles where a digit’s candidates cluster inside two boxes and the third box of the same band/stack is missing that digit on one line. The elimination you get is often a helpful local reduction that opens simpler solving techniques.

When to look for it

- Look when a single digit's remaining candidates in two boxes that share a band (same three rows) or a stack (same three columns) are confined to exactly two lines (two rows for a band, or two columns for a stack).
- Also look when the third box in that band/stack has no candidate for that digit on one of those two lines (that’s the “empty” part of the rectangle).
- It’s most effective after you’ve penciled basic candidates (singletons, box/line reductions, and naked pairs/triples). If you notice a digit concentrated in two boxes and forming an L or rectangle shape, pause and check the third box in that band/stack for a line with no candidates.

Step-by-step approach

1. Pick a digit and scan a band (three boxes in the same three rows) or a stack (three boxes in the same three columns).
2. If two boxes in that band/stack contain all the remaining candidates for that digit and those candidates sit on only two different lines (two rows in a band, two columns in a stack), mark those two lines.
3. Check the third box in the band/stack. If that third box has no candidates for the digit on one of those two marked lines, you have an empty rectangle for that digit.
4. Any cell that sees both candidate positions that form the “top” (or “left”) corners of the rectangle can have that digit eliminated, because if the digit were in that target cell it would force an impossible placement across the band/stack.

Clear candidate-grid example (walkthrough)

Look at the candidate diagram provided. We focus on a single digit — call it 7 for the example — and we’ll use the top three boxes (the top band). In the left and middle boxes of that band the only 7-candidates sit in rows 2 and 3 and specifically at the four marked positions. The right top box has no candidate 7 in row 2 — that is the box-line absence that makes the rectangle "empty."

Why those four marks matter: because the two boxes (left and middle) restrict 7 to only rows 2 and 3 inside those boxes. Those two rows form the opposite sides of a rectangle across the two boxes. The absence of any 7 in row 2 inside the right top box completes the empty rectangle condition.

What you can eliminate: any cell that simultaneously sees the two corner candidates that sit on the same row/column sides of the rectangle can’t be a 7. In the diagram we show a target cell (highlighted as an elimination) that sees both of those corner candidates; 7 can be removed there.

How to use this in your solve

- Pencil candidates for one digit at a time in a band/stack when you suspect concentration. Empty rectangle is not discovered by scanning all digits at once; it’s a targeted check.
- If you see candidates limited to two lines in two boxes, immediately check the third box for a line with zero candidates for that digit. If that absence exists, look for a cell that sees both corner candidates — that’s where you may have an elimination.
- Empty rectangle often appears after standard reductions have been applied. If you still have a lot of candidates penciled, simplify first; the pattern jumps out better when candidates are trimmed.

Practical tips and pitfalls

- Be sure the two boxes you use are in the same band (same three rows) or same stack (same three columns). The technique relies on that shared alignment.
- Don’t confuse an empty rectangle with a plain box-line reduction. Empty rectangle needs the combination of two boxes confining candidates to the same two lines and a box-line absence in the third box.
- Mark the two lines clearly (with pencil or a small mark) so you don’t lose track of which rows/columns are involved.
- This technique eliminates a single candidate at a time; it’s a local cut that may enable simpler eliminations afterward.

Summary

Empty rectangle is a compact, one-digit tactic that connects a box-line absence in one box to a pair of confined candidates in the adjacent boxes. It rewards careful candidate work in a band or stack and will often produce a single useful elimination that lets simpler methods finish the job. Use it whenever a digit looks concentrated in two boxes and the third box is missing that digit on one of the candidate lines.