Skip to content
Puzzle Quarry

Sticking a Block to Its Run

Now the filled cells start paying. A block you can see is a piece of some run, and knowing which runs it could be is enough to settle cells at both ends of it.

A block against a wall belongs to a run that starts there

The clean case is a filled block with a cross immediately to its left, or the edge of the board there. Whatever run that block belongs to must begin at the block’s first cell, because there is nowhere further left for it to start. So the run extends from there for as long as its number says, and if every candidate number is at least four, the next three cells are filled whichever candidate it turns out to be.

The same argument runs the other way from a wall on the right, and the two combine on a block walled at both ends: there the run is pinned exactly, its length is known, and the numbers can often be matched to it uniquely. A block walled on both sides whose length matches only one of the line’s numbers is the strongest single fact a nonogram line ever offers.

What the argument needs is a wall, which is why it sits above the pocket argument rather than below it. You cannot glue a block to anything until you know where the pockets end, and that is exactly the work the rung below does.

Before — a filled cell in the last column with a wall above it
1331111
1
31
3
2
1
After — the cell it forces is decided
1331111
1
31
3
2
1
The last column reads 1 1 and one of its cells is already filled with a proved blank directly above it. Whichever of the two runs that cell belongs to, the run starts there and is one cell long, so the cell below it is blank. The reasoning used the block and the wall beside it, and nothing else in the column.

what the reasoning read what it decides what it wrote

The block long enough to be its own wall

A filled block of three or more is worth reading even with nothing beside it, because its length already rules most numbers out. Only runs at least that long can contain it, and if the line has just one number that big then the block belongs to that number and its position is bounded on both sides. A block of one or two adrift in an open stretch does not carry enough to do this, which is what makes it the business of the rung above.

Three is where the line was drawn, and it was drawn by measurement rather than by taste. Treat every floating block as usable and the top rung of the ladder empties out; treat none of them as usable and the top rung swallows a third of every board generated and nearly every drawing in the library. Three leaves all four levels populated at every size on the site.

In practice the move you make most often is the small one: a block of three sitting under a number that says five, with a cross four cells to its left, so the two cells to its right go in. It is unglamorous and it is where hard boards give way.

What makes a board hard rather than long

A hard board here is one where the solver, always taking the cheapest available move, was at some point forced into this — no long number left to overlap, no line settled at its ends, no pocket to eliminate, and a block somewhere that had to be attached to a run before anything moved. That is a statement about the board containing a genuine bottleneck rather than about its size.

It is also why size and difficulty are separate choices here. A twenty at easy is four hundred cells and no bottleneck at all; a ten at hard is a hundred cells and at least one. Most people find the second takes longer, and nearly all of them find it more interesting.

Where this sits on the ladder

Common questions

What counts as a wall?

A crossed cell or the edge of the board. Either one stops a run, so a filled block sitting against one belongs to a run that begins exactly there.

Why is a block of three special?

Because its length alone rules most numbers out, so it can be used without a wall beside it. One or two filled cells adrift carry too little, and using them is the argument on the rung above this one.

How is this different from a filled cell at the start of a line?

That one needs no wall — the edge of the line is the wall, and it is an argument you can make while looking only at how the line begins. This one needs you to have found the walls first.

More Nonogram pages