Skyscrapers

What the border numbers really tell you

Each number outside the grid counts how many towers you would see looking down that row or column, given that a taller tower hides every shorter one behind it. Two clue types do your opening work for free. A clue of 1 means the first tower is the tallest on the board, so the maximum height goes right against that edge, blocking everything behind it. A clue equal to the grid size means you can see every tower, so the line climbs in strict order from that edge: 1, 2, 3 and up, the whole row written out for you. On a five-wide grid that generous clue is a 5; on a six-wide grid, a 6. Fill every one of those in before you touch anything else.

The counting rule

Here is the workhorse deduction for everything in between. A clue of k means the cell nearest that clue can be no taller than the grid size minus k plus one. Take a five-wide grid with a 3 at the left end of a row. To see three rooftops you need three towers, each taller than the last. If the first tower were a 4, only the 5 could ever rise above it, so you could never see more than two. So the first cell is at most 3. Run the same count one step in: the second cell is at most 4. Do this along every clued line before any fancy logic and you will be surprised how much of the grid is already pinned down. The rule also flips into a sanity check. If a line you have pencilled in would show the wrong number of rooftops, one of your earlier entries is wrong, and it is cheaper to find that now than three deductions later.

Working the middling clues

The clues between the extremes are where the puzzle lives. A 2 on a five-wide line says the tallest tower is not first, and exactly one tower shows before the 5 does, so the line starts with something midsized and then spikes. Each clue quietly bans certain heights from the near cells, and your job is to collect those bans.

• The tallest tower satisfies a clue of 1 only when it sits right on that edge, so use the 1s to place your maximum heights early.
• Underneath the theme it is a Latin square: every height appears once per row and once per column. Ordinary Sudoku-style elimination works right alongside the visibility logic, and you should bounce between the two constantly.
• Clues at opposite ends of a line watch the same towers from both sides. A check worth knowing: the two clues on one line can never add up to more than the grid size plus one. A big number on one side pushes the tall towers away from it, which feeds the clue at the far end.
• When a line is nearly full, read it as a running tallest-so-far from the clue side and count the rooftops. It is the fastest way to test an idea before you commit it.

Grid size changes the character of the puzzle more than the rules do. On a four-wide grid the extreme clues nearly solve the board on their own, while five-wide and up lean hard on the counting rule and patient elimination. If a big grid stalls, re-count the rooftops on every line you have partly filled. Nine times out of ten the mistake is already sitting on the board, quietly poisoning every deduction that came after it.

A skyline built from Latin squares

Strip away the theme and Skyscrapers is a Latin square, the same every-symbol-once-per-line backbone as Sudoku, dressed up with a line-of-sight rule. Latin squares are old mathematics: Leonhard Euler was studying them in the 1700s, long before anyone thought to stack them into city blocks. The visibility twist arrived with the modern puzzle boom, and Skyscrapers, sometimes printed under the name Towers, has been a fixture at the World Puzzle Championship for years, because it rewards clean chains of deduction rather than trial and error. There is always exactly one skyline that fits every clue. No guessing is ever needed, and the pleasure of the thing is watching a city rise one certain tower at a time.