Fish & Wings
Skyscraper
A Skyscraper removes one digit using two columns where that digit has exactly two places each. One end of each column sits in the same row, so those two cells cannot both hold the digit — and therefore at least one of the two other ends, the roofs, must. Any cell that sees both roofs can lose the digit. Swap rows and columns and the same argument works on two rows.
What you need to see on the grid
Look for a Skyscraper one digit at a time, once singles, locked candidates and subsets have run dry. It is the first pattern the app tries after them. Four terms carry the method:
- Strong link
- A column (or row) where the digit has exactly two places. One of the two must hold it, because the column needs the digit somewhere. A Skyscraper is built from two of them.
- The base
- One end of each link, the two lying in the same row. Two cells of one row never both hold the same digit, so at most one base cell does.
- The roofs
- The other end of each link. They must not share a row: if they did, the four cells would be an X-Wing, which clears the two rows instead.
- The row version
- Two rows with two places each, their base in the same column. The steps are identical with "row" and "column" exchanged.
How to apply it, step by step
One example runs through all four steps: the digit 3, late in an expert puzzle from the app, at the exact point where its hint needs a Skyscraper. 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 column where the digit has only two places
Pick a digit and read the columns one at a time. You want one where the digit is a candidate in exactly two cells. Column 6 gives 3 just R1C6 and R7C6: the column must hold a 3, so one of those two cells is the 3. That is a strong link.
Three places will not do — the column could then put its 3 in the third cell and promise nothing about the other two.
Only column 6 matters here. 3 is a candidate in exactly two of its cells: R1C6 and R7C6. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 6 R3 1 R4 7 R5 8 R6 5 R7 R8 2 R9 4 - Digit already on the grid
- Column where 3 has exactly two places
- A place for 3 in a link column
- Candidate 3, the digit studied
- Not needed for this step
Filled in: column 6. 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 second such column whose end shares a row with the first
Look for another column where the same digit has exactly two places, one of them in the same row as an end of the first link. Column 7 gives 3 the places R1C7 and R8C7, and R1C7 shares row 1 with R1C6.
Those two cells in row 1 are the base of the Skyscraper; the other two ends, R7C6 and R8C7, are its roofs. Check that the roofs are in different rows — here rows 7 and 8. Roofs in the same row would make an X-Wing instead.
Columns 6 and 7. Numbered in order: roof R7C6, base R1C6 and R1C7 in row 1, roof R8C7. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 6 R3 1 6 R4 7 R5 8 1 R6 5 R7 7 R8 2 R9 4 2 - Digit already on the grid
- Column where 3 has exactly two places
- A place for 3 in a link column
- Chain cell, numbered in order
- Candidate 3, the digit studied
- Not needed for this step
Filled in: columns 6 and 7. The other 63 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 from every cell that sees both roofs
R1C6 and R1C7 share row 1, so at most one of them is 3. If R1C6 is not 3, its link puts the 3 on the roof R7C6; if R1C7 is not 3, its link puts the 3 on the roof R8C7. One of the two base cells is certainly not 3 — so at least one roof is.
A cell that sees both roofs therefore cannot be 3. R7C9 sees R7C6 along row 7 and R8C7 inside box 9; R8C5 sees R8C7 along row 8 and R7C6 inside box 8. Both lose 3. The other cells seeing both roofs, R7C7, R7C8, R8C4 and R8C6, are filled or do not list 3. A cell that sees only one roof keeps its 3: the other roof may be the one holding it.
Technique: Skyscraper
The two link columns and every cell that sees both roofs R7C6 and R8C7. 3 leaves R7C9 and R8C5. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 6 R3 1 6 R4 7 R5 8 1 R6 5 R7 7 R8 7 2 R9 4 2 - Digit already on the grid
- Column where 3 has exactly two places
- A place for 3 in a link column
- Chain cell, numbered in order
- Candidate 3, the digit studied
- Candidate removed
- Not needed for this step
Filled in: columns 6 and 7; cells R7C8, R7C9, R8C4 and R8C5. The other 59 cells are left blank: they are not needed for this step. Removed: 3 from R7C9 and R8C5.
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Step 4Collect what the elimination unlocks
A Skyscraper never places a digit by itself, so look at the cells it just cleared. R8C5 had two candidates, 3 and 5. With 3 gone, only 5 is left: a Naked Single, and the app's next hint in this puzzle.
Check every cell that lost a candidate, and the rows, columns and boxes around it, before you move on.
Technique: Naked Single
Row 8 after the Skyscraper. R8C5 has only 5 left. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 R6 R7 R8 6 1 9 7 5 2 4 R9 - Digit already on the grid
- The cell that must be 5
- Candidate 5, the digit studied
- Digit placed
- Not needed for this step
Filled in: row 8. The other 72 cells are left blank: they are not needed for this step. Placed: R8C5 = 5.
Common mistakes
Building on a column with three places
Both towers must be strong links: exactly two places each. With a third place, the column can hold its digit there, both base and roof can be empty of it, and nothing follows. Count the places in all nine cells of the column, not only the ones near the pattern.
Clearing cells that see only one roof
The pattern proves that one roof holds the digit, never which one. R7C5 sees the roof R7C6 along row 7 but not the roof R8C7: if R8C7 is the 3, nothing stops R7C5 from being 3 as well, so it keeps its candidate. Only cells seeing both roofs lose it.
Mistaking an X-Wing for a Skyscraper
When the roofs line up in the same row as well, the four cells form an X-Wing: the digit is then removed from the rest of the two rows, not from cells seeing both roofs. Check the roofs' rows before you decide which pattern you have.
Recent daily puzzles that needed this technique
Keep going
Learn the move, not just the digit
In a real game, when a Skyscraper 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.