Fill every cell with a building height from 1 through the size of the grid.
Each height appears exactly once in every row and exactly once in every column.
A number outside the grid says how many buildings are visible when looking along that row or column from that side. A building is visible if it is taller than every building before it in that view; a taller building hides all shorter buildings behind it.
Respect every shown border clue. For example, viewing 2, 4, 1, 3 from the left shows two buildings: the 2 and the 4.
The puzzle is solved only when the entire grid satisfies all row, column, given-building, and shown-clue rules. Every published puzzle has exactly one valid completed grid.
If only one building is visible from a side, the very first cell must be the tallest height on the board — nothing else could hide every building behind it.
Example: On a 5×5 board a left clue of 1 puts a 5 in the first cell of that row.
When the clue equals the size of the grid, every building has to be visible, which only happens if the heights climb one at a time from that edge.
Example: On a 5×5 board a top clue of 5 fills that column downwards with 1, 2, 3, 4, 5.
A clue of k needs k − 1 shorter buildings in front of the tallest one, so the tallest cannot sit in any of the first k − 1 cells. The same bound rules the second-tallest out of the first k − 2 cells, and so on.
Example: A left clue of 3 on a 5×5 row means the 5 is not in cell 1 or cell 2 of that row.
Opposing clues both describe the same line, and their sum is bounded: a line of N cells can show at most N + 1 buildings in total from both ends. A small pair therefore pins the tallest building near the middle.
Example: On a 5×5 row clued 2 on the left and 2 on the right, the 5 has to sit where exactly one building hides on each side.
A clue plus the distinct-heights rule usually leaves only a handful of possible arrangements for a line. Write them out and keep only the positions every one of them agrees on.
Example: On a 4×4 board a clue of 2 allows only lines starting 3, 4 or 2 with the 4 second, or 1 with the 4 later — whatever they share is certain.
Every height you place removes that height from its row and from its column at once. After filling a cell, walk both lines it sits on before looking anywhere else.
Example: Placing a 4 in row 2 means no other cell in row 2, and no other cell in that column, can be a 4.
Once a line is full and its clue turns green, it is finished. Treat its heights as fixed facts for the crossing lines rather than revisiting it.
Example: A completed column with a green top clue supplies one certain height to each of the rows it crosses.
When two heights are both possible in a cell, try one and follow it out. If it forces a line that no arrangement can satisfy, its clue turns red and the other height is the answer.
Example: Pencil in a candidate as a note. If a clue on its row or column goes red, undo it and place the other height for certain.