Where Can You Use a Skyscraper When Locked Candidates Stop Helping?

Look beyond locked candidates to two rows or columns

When all candidates for a number in one box lie in a single row or column, you can remove that number from the rest of that row or column outside the box. This is Pointing. Conversely, when all candidates for a number in a row or column lie in one box, you can remove that number from the rest of the box outside that row or column. This is Claiming. Together, these two methods are called locked candidates.

If these methods leave nothing more to remove, look for two rows or two columns, each with exactly two possible cells for the same number. Skyscraper connects candidate positions across two rows or columns. Its starting point differs from locked candidates, where you look for candidates confined to one box.

Conditions — one aligned base and two offset tops

For a pattern based on rows, all the following conditions must hold. For a pattern based on columns, swap rows and columns.

1. The same number can go in exactly two cells in each of two rows. This does not mean that each cell must have only two candidates in total.
2. One candidate cell from each row lies in the same column. These two cells form the base.
3. The other two cells, the tops, lie in different columns.
4. Another cell that sees both tops must contain the same candidate for there to be anything to remove. Two cells “see” each other when they share a row, column or box.

The two base cells share a column, so they cannot both contain the number. At least one base cell must therefore not contain it, and the top in that row must contain it. Since at least one of the two tops contains the number, you can remove it from any cell that sees both.

Example — cells that see both tops for 6

c1c2c3c4c5c6c7c8c9

r1r2r3r4r5r6r7r8r9

5

1

6

8

67

8

3

5

2

9

1

36

2

8

4

5

7

36

56

3

7

4

1

8

2

9

56

9

8

6

2

3

7

4

146

9

7

5

3

8

8

6

9

7

3

2

5

4

5

2

8

9

1

6

3

7

7

5

4

6

8

2

9

BBTT

• In row 3, 6 can go only in r3c3 · r3c9.
• In row 4, 6 can go only in r4c1 · r4c9.
• The base cells r3c9 · r4c9 meet in column 9, while the tops r3c3 · r4c1 lie in different columns.

r2c1 sees r3c3 through box 1 and r4c1 through column 1. r6c3 sees r3c3 through column 3 and r4c1 through box 4. Both cells see both tops, so remove 6 from them. In r2c1, candidates 6 and 7 leave only 7; in r6c3, candidates 1, 4 and 6 leave 1 and 4. This step itself removes candidates; placing a number is the next step.

When the pattern fails — similar lines do not work with three candidate cells

The board below is the example board with the 9 in r3c1 removed. You can draw the same lines between the original four cells, but 6 now has three possible cells in row 3: r3c1 · r3c3 · r3c9.

c1c2c3c4c5c6c7c8c9

r1r2r3r4r5r6r7r8r9

2379

24

5

1

6

79

49

8

34

679

8

46

3

5

79

149

16

2

369

1

36

2

8

4

5

7

36

56

3

7

4

1

8

2

9

56

15

9

8

6

2

3

7

4

15

126

24

146

9

7

5

3

16

8

8

6

9

7

3

2

14

5

14

4

5

2

8

9

1

6

3

7

13

7

13

5

4

6

8

2

9

!TBTB

Even if r3c9 is not 6, you cannot conclude that r3c3 must be 6, because r3c1 is another possible position. The first thing to find on this board is that r3c1 is the only place for 9 in row 3. Only after filling that number do the example's Skyscraper conditions hold again.

When you cannot use it, and common mistakes

• The base is not aligned. If the two base cells are not in the same column, both might be 6, so you cannot draw a conclusion about the tops.
• You pick two candidate cells and draw a line. Always check that there is no third candidate cell in the same row.
• You remove a candidate from a cell that sees only one top. You do not yet know which top contains the number, so the cell must see both.
• You fill both tops. At least one top contains the number; this does not mean that both do.

Practice puzzle

c1c2c3c4c5c6c7c8c9

r1r2r3r4r5r6r7r8r9

3

6

8

1

7

2

4

9

5

5

9

4

3

8

6

2

7

2

1

5

4

9

6

8

3

4

3

2

8

157

6

9

8

6

4

9

157

3

157

2

9

5

6

2

3

8

4

2

8

7

9

3

15

15

4

6

6

4

3

2

8

9

157

1

5

9

7

6

4

2

3

8

BBTT

Which candidate can you remove from r4c7 and r8c8? Count the possible positions for 5 in rows 5 and 7, then check which column contains the base.

Show answer

Remove 5 from both cells. In row 5, 5 can go only in r5c6 · r5c8; in row 7, only in r7c6 · r7c7. The base lies in column 6, and the tops are r5c8 · r7c7. r4c7 sees the first top through box 6 and the second through column 7. r8c8 sees the first through column 8 and the second through box 9. Removing 5 from candidates 1, 5 and 7 in both cells leaves 1 and 7.

Further reading

First read How to Use Sudoku Candidate Notes and Stuck on Sudoku? A 7-Step Checklist. Next, read How Does XY-Wing Differ from X-Wing, and Where Can You Remove Candidates?.