The Skyscraper Technique in Sudoku

The shape of a Skyscraper

Take one digit and find two rows in which it has exactly two candidate cells each. If one end of both pairs sits in the same column, that shared column becomes the base of the skyscraper. The other two candidate cells, which lie in two different columns, are the tops of the two towers, and the name paints the picture nicely: two towers rising from a common base. The pattern works just as well with the roles of rows and columns swapped, so it is always worth searching in both directions. The logic is short. The base column can contain the digit in only one of its two cells, since a column holds each digit exactly once. Whichever base cell turns out to be false forces its own row to place the digit in that row other candidate, which is the top of that tower. So if the left base is false the left top is the digit, and if the right base is false the right top is the digit. One base must be false, therefore at least one of the two tops is certainly the digit, and that guarantee is what the elimination rests on.

A worked example

Let the digit be four. Say row two has four only in r2c3 and r2c7, and row six has four only in r6c3 and r6c9. Both pairs share column three, so r2c3 and r6c3 are the base and column three is the shared column that anchors the pattern. The tops are r2c7 and r6c9, sitting in two different columns, seven and nine. Because at least one of those two tops must be four, any cell that can see both of them is forbidden from being four itself. Here r2c9 sees r2c7 along row two and sees r6c9 down column nine, so it qualifies, and r6c7 sees r6c9 along row six and r2c7 down column seven, so it qualifies too. You therefore strike four from both r2c9 and r6c7. Notice how little you needed to know: just two rows where the digit was pinned to two cells, sharing a column at one end. That modest requirement is exactly why skyscrapers appear so often once you start looking for them.

How to use it

1. Choose a digit and mark every row where it has only two candidate cells.
2. Find two such rows whose pairs share a column at one end, which becomes the base.
3. Identify the two tops, the candidate cells that are not in the shared column.
4. Erase the digit from every cell that sees both tops.
5. Run the whole search again with columns in place of rows, since a skyscraper can stand either way up.

Where the eliminations hide

The cells that see both tops are usually a small handful, so the skyscraper is an efficient, surgical tool rather than a sweeping one. Most of the time the useful victims sit in the two rows or columns that contain the tops, or in a box that one top happens to share with the other tower line. It is worth checking each box the tops touch, because when the geometry lines up neatly a single skyscraper can clear two or three candidates at once, and occasionally one of those removals collapses a cell to a single value and cracks the board open. If you find a valid skyscraper but it removes nothing, that is normal and not a failure of the pattern; it simply means no cell currently sees both tops. Note it and move on to another digit, then come back later, because as the grid fills the very same skyscraper may start to bite.

A close cousin, spotting it faster

The Skyscraper is a near relative of the Two-String Kite and the X-Wing, and all three chain two conjugate pairs of a single digit before removing that digit from cells seeing both loose ends. The X-Wing lines its pairs up into a perfect rectangle, the kite bends the shape around a shared box, and the skyscraper offsets the two tops into different columns. Learn the skyscraper first, because its outline is the clearest of the three, and the others will feel like small variations on a shape you already know. As for finding them quickly, work one digit at a time and look only at rows or columns where that digit is already pinned to two cells; full notes make those pairs jump straight out. In Shapedoku, where each value is a distinct glowing shape, a row with just two candidate cells for one shape is easy to pick out from the board, which turns the skyscraper from a puzzle-hunter trick into a move you can genuinely spot at a glance while you play.