Rectangular nonograms: why 15×10 is not 15×15

The arithmetic, in one line

Everything here follows from a single rule. A clue of length k in a line of n cells forces 2k − n cells to be filled, wherever the block ends up sitting. If that number is zero or negative, the clue tells you nothing on its own; if it is large, the clue hands you a block of certain cells for free.

The important part is that n is the length of the line, not the size of the grid. In a 15×10 board the rows run across the width and are fifteen cells long, while the columns run down the height and are ten. So a clue of 8 in a column forces six cells, and the same clue of 8 in a row forces one. Identical number, six times the payoff, purely because of which direction it points.

That is the whole idea. On a rectangle the shorter axis is where a typical clue converts into certainty, so that is where the first pass belongs. The longer axis is where the biggest single blocks appear when a clue happens to be large — a clue of 13 in a fifteen-cell row forces eleven cells — but those are rarer, and waiting for one is a bad opening.

Reading the shape before you start

Grids on this site are written width by height, so the first number is how wide and the second is how tall. A 15×10 is a landscape board: wide rows, short columns, so the column clues pay out first. A 10×15 is portrait: short rows, tall columns, so it is the row clues that go first. The two look almost the same in a listing and behave in opposite ways.

It is worth making this an explicit habit rather than something you notice halfway through. Before placing anything, ask which axis is shorter, and start there. Most people open at the top-left corner and work along a row out of reading habit, which on a portrait board is exactly the wrong axis and quietly costs the first ten minutes.

The extreme case in the catalog is the single 5×15, a key drawn on a board three times taller than it is wide. Rows five cells long, columns fifteen. Almost every useful overlap lives on the rows, and the columns are long enough that most of their clues force nothing at all until the rows have pinned things down. It is the clearest demonstration of the principle available, precisely because it is so lopsided.

Where the rectangles are

Twenty-four of the catalog's one hundred and sixty-two boards are non-square — about one in seven — spread across seven shapes. The two commonest are 10×15 with eight boards and 15×10 with seven, which is convenient: they are the same dimensions rotated, so solving one of each is a direct comparison of the two openings.

The rest thin out quickly: three 15×20s, three 20×10s, and a single board each at 10×20, 20×15 and 5×15. Eleven of the twenty-four are wide and thirteen are tall, which is nearly a balanced split overall — but that balance disappears completely once you look at where they sit.

The subject decides the shape

Animals has seven rectangular boards and every single one of them is wide. Not one is taller than it is broad. That is not a decision anybody made about grids — it is a fact about animals. A horse, a fish, a swan, a bat, a duck seen side-on is a fundamentally wide subject, and drawing one on a square grid wastes the top and bottom of the board. Dog, Fox and Swan are 15×10; Bat, Cat and Duck are 20×10; Fish is 20×15.

Food and drink runs the other way. It has eight rectangular boards, the most of any category, and six of the eight are tall. Bottles, cups, cakes and cans stand up. The two categories between them account for fifteen of the twenty-four rectangles on the site and pull in exactly opposite directions, which makes them the natural pair to practise on: work an animal to learn the landscape opening, then a drink to learn the portrait one.

Three categories have no rectangles at all. Zodiac signs is twelve boards all at 15×15, letters and numbers is thirty-five boards all at 5×5, and hats and masks has none either — a hat is roughly as tall as it is wide, and a glyph is designed to sit in a square. If you want to practise lopsided grids, those three have nothing for you, and that is a property of their subjects rather than an oversight.

What changes in practice

Open on the short axis and finish the pass before switching. The temptation on a wide board is to jump to a long row the moment a big clue appears, but a complete pass down the short columns usually leaves enough anchors that the long rows resolve almost on their own afterwards.

Expect the long axis to be where you mark empty cells rather than fill them. On a fifteen-cell row with clues of 2 1 3, the useful early deduction is rarely which cells are filled — it is which stretches cannot contain a block at all. On a rectangle that work concentrates on the long side, and skipping it is what makes lopsided boards feel like they stall.

Finally, do not read a rectangle as easier than a square of the same larger dimension. A 20×10 has two hundred cells against a 20×20's four hundred, but the twenty-cell rows are exactly as unhelpful in both — the overlap rule needs a clue of eleven before it forces even two cells at that length. What the shorter columns buy you is a way in, not a smaller problem.

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