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Puzzle Quarry

How to Solve a Skyscrapers Puzzle

One real 5×5 with seven numbers round it, from a blank plot to a finished city. Every move below is a move the solver made, in the order it made it.

The order that finishes boards

Read every printed number once before writing anything. A 1 puts the tallest tower against that edge outright. A number equal to the side lays the whole street out as a staircase. Everything in between caps the lots nearest that edge, and those caps are free — they cost nothing to work out and they never have to be revisited, because the rim never moves.

Then, and only then, start crossing heights out. Every height you settle removes itself from the rest of its street and the rest of its avenue, and the value of doing that bookkeeping honestly is that it makes the next round of view arithmetic much shorter. Solvers who skip it end up re-deriving the same three eliminations four times and calling the puzzle tedious.

Writing out the orderings of a whole street is a slower move and belongs last. It is genuinely worth doing when a run of free sweeps has produced nothing, because it uses the one thing no cheaper move uses — the order of the heights rather than the set of them — but reaching for it early means writing out arrangements that a free sweep was about to make irrelevant.

One board, six arguments

The board below is a 5×5 with seven numbers round the rim, graded expert — chosen not because a beginner should start there but because its opening happens to use all six arguments within twenty-eight moves, which no gentler board does. The six positions shown are the six moments where something new is being used; the thirty-nine moves in between are more of the same.

They do not arrive in ladder order, and that is worth expecting rather than being surprised by. The solver always takes the cheapest move available at that instant, so the sequence here runs first rung, second, fourth, fifth, third, sixth — the third-rung argument becomes available late on this board because the column it lives in stays wide open until two of its neighbours have been settled.

Read each pair of grids left to right. On the left is the position as it stood, with the numbers and lots the argument read marked and the lot it is about to decide outlined. On the right is the same position a moment later. Small digits in a lot are the heights still standing there; watching one disappear is very often the whole content of a move, which is why they are drawn rather than left out.

  1. Start at the rim, before anything is written. The 2 facing the top row from the right says two roofs are in sight from that side, so the lot against that edge cannot hold the tallest tower — a five standing there would hide the whole row and the view would be one. The 5 goes, and nothing on the board had to be read to know it.

    The position
    32
    5
    2
    3
    22
    A moment later
    32
    5
    2
    3
    22

    Reading the rim

  2. The 5 on the left of the third row has already laid that row out as a staircase, one storey at a time, which puts a single-storey building in the corner. That 1 now leaves every other lot in its column and every other lot in its row — eight eliminations from one settled tower, and this is the move that does most of the work on every board here.

    The position
    32
    51
    2
    3
    22
    A moment later
    32
    512
    2
    3
    22

    Singles

  3. Nothing free is left, so a tower gets tried. The 2 under the last column means two roofs are in sight looking up it, and it already passes a five in the middle row. A four standing immediately below that five would be a second roof before the five was even reached, making three in all, so the four cannot go there.

    The position
    32
    512345
    2
    3
    22
    A moment later
    32
    512345
    2
    3
    22

    Blocked heights

  4. The fourth row carries a 2 at its right-hand end and now has enough settled around it to be written out properly. Of the orderings that still show exactly two roofs from that side, none puts a 3 in the fourth lot and none puts a 5 in the first — two eliminations, neither of them visible without the list.

    The position
    32
    512345
    2
    3
    22
    A moment later
    32
    512345
    2
    3
    22

    Counting the line

  5. Back to a free sweep, and the second column has quietly run out of room. The tallest tower has been crossed out of four of its five lots by the moves above, so it stands in the one that is left — even though that lot still carried four candidates a moment ago and announced nothing at all.

    The position
    32
    512345
    2
    3
    22
    A moment later
    32
    5
    512345
    2
    3
    22

    Only one home

  6. The last and hardest rung. The top row has a 3 against its left edge and a 2 against its right, and neither number alone has anything further to say about its fourth lot. The orderings that produce three roofs one way and two the other are a much shorter list, and every single one of them puts either the shortest tower or the tallest in that lot.

    The position
    32
    5
    512345
    2
    53
    22
    A moment later
    32
    5
    512345
    2
    53
    22

    Both ends at once

what the reasoning read what it decides what it wrote

Thirty-nine more moves of the same six ideas, and every lot holds one tower. Here is the finished city.

The finished skyline
3235142
45231
512345
314522
541233
22

What to do with a stalled board

Re-read the numbers nearest whatever you last wrote. By far the commonest cause of a stall at this level is a rim number you spent on the first pass and have not looked at since a height went in halfway along its street. A view is not a one-time calculation; every height settled anywhere along a line shortens the list of orderings that line still allows.

If that fails, take one street and write its orderings out properly. Choose the shortest list rather than the longest street — a line with two settled lots and a number at one end usually has three or four arrangements left, and what all of them agree on is a free elimination that no amount of staring would have produced.

And if a full sweep and a written-out street both come up empty, look at the pair of numbers facing each other across a line and ask what they say together. What the board never wants is a guess: every skyline published here was checked to be finishable without one, so a board that will not move is a board with something still on it to find.

Common questions

What is the first move in a skyscrapers puzzle?

Read the rim. Put the tallest tower next to every 1, lay a staircase along every street whose number equals the side, and cap the lots nearest every other number before writing anything else.

Should I write candidate heights in the lots?

Yes, from the second move onwards. Almost every argument above the first rung works by removing a height rather than settling a lot, and a board with nowhere to record that makes you derive the same elimination repeatedly.

Why does the board keep stopping dead near the end?

Almost always because a rim number spent on the first pass has not been read again. The orderings a street still allows shrink every time a tower is settled along it, and the shorter list is usually what the board was waiting for.

More Skyscrapers pages