Visibility: count what you can see
When the edge rules run out, read a line from its clue. Keep only the orders that show the right number of buildings, and strike any height none of them allow.
How it works
A line with a visibility clue can only be filled in certain orders. Write the heights each cell can still take, then ask which orders fit those and show exactly the clue’s count. A height that no fitting order puts in a cell is out. That is elimination.
You rarely need to list every order. Start from the tallest building: it is always seen, and everything after it is hidden. So the clue only counts the new highs up to the tallest.
When a line has clues at both ends, both must hold at once. Each clue alone may allow many orders, but together they often pin the tallest building to one cell.
A quick way to test orders
Read from the clue. Each time a building is taller than every one before it, count it. Stop at the tallest. If the count doesn’t match the clue, the order is out.
For hard lines, fix the tallest building’s cell first, then ask how many new highs the cells before it can make. Few cells before it means few buildings seen.
Why it is a proof
Every order that breaks the clue is ruled out, and what is left must contain the answer. A height that none of the survivors use in a cell can’t be the answer there. It is the same move as testing every case in case analysis.
Worth memorizing
The tallest building is always seen.
Everything behind it is hidden.
Keep only orders that show the clue’s count.
Strike heights no surviving order uses.
Two clues on one line must both hold.
Together they can pin the tallest building.
Practice
0 of 4 rightRow 1 has its 5 at the left end and shows 2 from the right. What height is in the ringed cell?
Row 4 shows 2 from both ends. Which cells can hold its 4?
Row 1 shows 3 from both ends. Where is its 5?
Row 2 shows 3 from the left and 2 from the right. What height is in the ringed cell?