Placing the Tallest Tower
A technique for Skyscraper Sudoku 5×5.
In skyscraper sudoku, a clue tells you how many towers are visible from its side. Every visible tower is taller than everything in front of it, and the tallest tower, the 5, is always among them. So if a clue is k, there have to be k-1 visible towers in front of the 5, and they each need a cell. The 5 can't stand in any of the k-1 cells nearest the clue.
This is the most useful rule in skyscraper after the edge clues, and it applies to every clue from 2 up.
The rule
| Clue | The 5 can't stand in |
|---|---|
| 2 | the nearest cell |
| 3 | the nearest 2 cells |
| 4 | the nearest 3 cells |
| 5 | the nearest 4 cells |
So a clue of 5 puts the 5 in the far cell. A clue of 1 is the opposite case: the 5 has to be in the nearest cell.
An example
Below the fourth column there's a 4. Four towers must be visible from the bottom, so three must stand in front of the 5. That keeps the 5 out of the bottom three cells of the column.
Two of those three already couldn't be 5: the fourth row has its 5 in the last column, and the bottom row has its 5 in the first. The middle cell still had 5 as a candidate, and now loses it.
That's worth following up. The top cell of the column can't be 5 either, because the top row's 5 is already placed, in the second column. So the only cell left for this column's 5 is the second one from the top. One clue, and the column's 5 is placed.
The app's Next step hint calls this "Skyscraper max-position bound".
Going further
The same reasoning applies to every tower, not just the 5. A tower at a given cell can only be so tall before it hides too much behind it. Counting cells from the clue (the nearest is 1), on a 5×5 board with clue k:
- the tower in cell i can be at most 5 - k + i.
So with a clue of 4, the nearest cell can be at most 2, the second at most 3, the third at most 4. With a clue of 3, the nearest is at most 3. It's the same idea: a tall tower near the clue doesn't leave room for enough visible towers. The app's hints only use this for the 5, but the general bound is just as sound and often settles a line sooner.
When neither bound decides anything, the method that always works is to check every ordering the line allows.
For the full rules, see the skyscraper sudoku guide.
More techniques
- Skyscraper Edge Clues: 1 and N: A clue of 1 puts the tallest tower right beside it. A clue equal to the line's length means the towers climb one step at a time.
- Checking Every Ordering: When the shortcuts run out, try each ordering of a line that the candidates allow, and keep only the digits that some valid ordering uses.