Empty Rectangle: A Box Cross Meets a Line

The Empty Rectangle rounds out the trio that started with Skyscraper and 2-String Kite. It's the hardest of the three to picture, because the name refers to an empty corner, not an empty box — that single mix-up is where most people get stuck on this one.

What is an Empty Rectangle?

In one box, every spot for a digit lies along a single row and a single column of that box — a cross, leaving an "empty rectangle" in the opposite corner where the digit has no candidates at all. Add a line outside the box where the digit fits in only two cells, one of them on an arm of the cross.

If that outside cell is the digit, it clears one arm of the cross, pushing the box's digit onto the other arm. If the other end of the line is the digit, it clears its own row or column directly. Either way, the cell where the line's far end meets the cross's other arm can't be the digit.

Why it works

The box needs the digit somewhere, and every option sits on the cross's row or column. The outside line needs it too, and has exactly two options. Trace both branches — whichever cell on the line turns out to be the digit, it always clears the same corner cell, so that cell can never be it.

A worked example

Here's the exact board the app uses for its own Empty Rectangle lesson:

Box/Line Reduction and singles do most of the work:

When they run dry, look at digit 4 in the top-right box. Its candidates form a cross along row 1 and column 9 — the opposite corner is the "empty rectangle." Outside the box, column 2 has a 4 in only two cells, and one of them sits right on the cross's row-1 arm:

Blue & violet: the cells the logic runs on · Rose: where the elimination lands

Follow it through: if R1C2 is the 4, row 1's arm of the cross is cleared, so the box's 4 must be on column 9's arm — specifically R2C9, since R1C9 is now taken by row 1. If R8C2 is the 4 instead, that directly rules out a 4 anywhere else in row 8. Either way, R8C9 — which sits in column 9 and row 8 — can't be a 4.

How to spot one

1. Look for a box where a digit's candidates form an L or cross shape, with one whole corner of the box empty of that digit.
2. Note which row and which column of the box the cross runs along.
3. Find a line outside the box where the digit fits in exactly two cells, one of them lined up with the cross.
4. Eliminate the digit from the cell where the line's other end meets the cross's other arm.