Fish & Wings
Empty Rectangle
An Empty Rectangle combines a box and a strong link. In the box, the digit's places all lie on one row and one column — the other four cells, the "empty rectangle", cannot hold it. A row outside the box has exactly two places for the digit, one of them in the box's column. Whichever way that row goes, the cell where its other end's column meets the box's row cannot hold the digit.
What you need to see on the grid
Look for an Empty Rectangle one digit at a time, when a box has only a handful of places left for it. Four terms carry the method:
- The cross
- One row and one column of the box that, between them, hold every place the digit has in the box — with at least one place on each arm away from where they meet.
- The empty rectangle
- The four cells of the box off the cross. None of them can hold the digit, which is what gives the pattern its name.
- Strong link
- A row (or column) outside the box where the digit has exactly two places: one on the box's column (or row), one beyond the box's stack (or band).
- The target
- The cell in the box's row and the column of the link's far end. It loses the digit.
How to apply it, step by step
One example runs through all four steps: the digit 8, in a hard puzzle from the app, at the exact point where its hint needs an Empty Rectangle. Every diagram shows the whole grid. The cells a step needs are filled in; the others are left blank and hatched, so you know you can ignore them. A digit missing from a drawn cell is already placed in its row, column or box, drawn or not.
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Step 1Find a box where the digit sits on one row and one column
Pick a digit and look at its places inside each box. You want a box where they all fit on one row plus one column of the box, with a place on each arm.
Box 1 gives 8 the places R2C2, R2C3 and R3C3: all on row 2 or column 3. R1C1, R1C2, R3C1 and R3C2 — the empty rectangle — are filled or do not list 8. So box 1's 8 is either in row 2 or in column 3.
Only box 1 matters here. 8 fits R2C2, R2C3 and R3C3 — the cross of row 2 and column 3. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 2 3 R2 R3 4 5 R4 R5 R6 R7 R8 R9 - Digit already on the grid
- Box being read
- A place for 8
- Candidate 8, the digit studied
- Not needed for this step
Filled in: box 1. The other 72 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.
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Step 2Find a strong link with one end on the box's column or row
Follow column 3 out of the box and look for a row where 8 has exactly two places, one of them in column 3. Row 6 gives 8 just R6C3 and R6C6, so one of them is the 8.
The other end, R6C6, must lie outside the box's stack — columns 1 to 3 — and it does. Row 6 must lie outside the box's band, rows 1 to 3, and it does too.
Column 3 runs from box 1 down to row 6, where 8 has only R6C3 and R6C6. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 2 3 R2 R3 4 5 R4 R5 4 R6 2 9 4 7 1 3 6 R7 2 R8 R9 6 - Digit already on the grid
- Box 1 and the link row
- A place for 8 in the box or the link
- Candidate 8, the digit studied
- Not needed for this step
Filled in: row 6; column 3; box 1. The other 58 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.
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Step 3Remove the digit where the link's far end meets the box's line
Row 6 has two options. If R6C3 is 8, column 3 has its 8, so box 1 cannot use R2C3 or R3C3 and must put its 8 at R2C2 — in row 2. If R6C6 is 8 instead, column 6 has its 8.
Either way R2C6 cannot be 8: it shares row 2 with R2C2 and column 6 with R6C6. 8 leaves R2C6.
Technique: Empty Rectangle
Box 1, the link row 6 and the target R2C6, on row 2 and column 6. 8 leaves R2C6. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 2 3 R2 R3 4 5 R4 R5 R6 2 9 4 7 1 3 6 R7 R8 R9 - Digit already on the grid
- Box 1 and the link row
- A place for 8 in the box or the link
- Candidate 8, the digit studied
- Candidate removed
- Not needed for this step
Filled in: row 6; box 1; cell R2C6. The other 62 cells are left blank: they are not needed for this step. Removed: 8 from R2C6.
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Step 4Collect what the elimination unlocks
An Empty Rectangle never places a digit by itself, so look around the cell it just cleared. Column 6 had two places for 8, R2C6 and R6C6. With R2C6 gone, R6C6 is the only one left: a Hidden Single, and the app's next hint in this puzzle.
Technique: Hidden Single
Column 6 after the Empty Rectangle. 8 now fits only R6C6. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 4 R2 R3 1 R4 2 R5 6 R6 8 R7 R8 3 R9 9 - Digit already on the grid
- Column 6: one place left for 8
- The cell that must be 8
- Candidate 8, the digit studied
- Digit placed
- Not needed for this step
Filled in: column 6. The other 72 cells are left blank: they are not needed for this step. Placed: R6C6 = 8.
Common mistakes
A box with a place off the cross
If even one place of the digit lies off the row and column you picked, the box can put its digit there, and the argument "if not in column 3, then in row 2" breaks. Check all four cells of the empty rectangle before you go on.
A link that is not strong
The row outside the box must leave the digit exactly two places. With a third place, R6C3 and R6C6 could both be empty of 8 and nothing would reach R2C6.
Clearing the wrong crossing
The target sits on the box's row — the arm the link does not touch — in the column of the link's far end: here R2C6. Other cells of column 6 see R6C6 but nothing in box 1's row, and cells of column 3 are the arm the link already uses; the pattern says nothing about either, so they keep their candidates.
Keep going
- Turbot FishTwo strong links on one digit, joined end to end: cells that see both free ends lose the digit.
- X-WingTwo rows lock one digit into the corners of a rectangle — every other cell of its two columns loses it.
Learn the move, not just the digit
In a real game, when an Empty Rectangle is what opens the next cell, the hint in the app names it and shows the cell it unlocks — the same move as step 4 above.