Lanterns strategy: numbers, light and dark cells

Play Lanterns

Lanterns rewards two habits: reading the numbers, and asking of every dark cell where its light could still come from. Almost every step in the puzzle is one or the other.

How the difficulty works

Each tier is set by the hardest step you'll need:

Easy (7×7) needs only the basics: numbers that are full or need every open neighbour, light that rules cells out, and dark cells with one way left to be lit.
Medium (9×9) needs at least one lantern that would block: a cell where a lantern would leave a number short, or a dark cell with nothing to light it.
Hard (10×10) needs a short what-if: suppose a lantern here, follow the basics a few steps, and watch something break.

The basics

Light rules out. Every cell a lantern lights is closed to other lanterns. Dot them in your head, or just read the shading.
Full numbers. A number with all its lanterns has no room for more: its other neighbours are empty.
Hungry numbers. A number with exactly as many open neighbours as lanterns it still needs: they're all lanterns.
One way to be lit. A dark cell can only be lit from itself or from a cell it can see. When only one of those is still open, the lantern goes there.

Zeros

A 0 is the easiest number of all: none of its neighbours holds a lantern. Dot them straight away — they're often exactly the cells that make a nearby dark cell run out of options.

The 0 at row 4, column 7 (outlined) has no lanterns beside it, so all its open neighbours are crossed out. From Easy puzzle #2 — play it

A cell that must light itself

Sometimes every cell a dark cell can see is already ruled out. Then the only lantern that can light it is one in the cell itself.

Everything row 1, column 7 can see along its row and column is ruled out, so it needs a lantern of its own. From Easy puzzle #1 — play it

Medium: a lantern that would block

Before placing a lantern, look at what it would light. Every one of those cells is ruled out at once — and if that takes away the last chances of a number or a dark cell, the lantern can't go there.

A lantern at row 4, column 8 (crossed) would light the tinted cells, and with them the last open cells beside the 1 at row 5, column 9 (outlined). That number would be left short, so the cell is empty. From Medium puzzle #1 — play it

A lantern at row 1, column 9 (crossed) would rule out every cell that could still light row 3, column 8 (outlined), without lighting it. So the cell is empty. From Medium puzzle #5 — play it

The most common case is a cell diagonal to a number. A lantern there lights the two neighbours of the number it touches at once. So around a 3 with four open sides all four diagonal cells are empty, and around a 2 with three open sides, so are the two diagonals between its open sides.

Hard: the what-if

When nothing else works, pick a cell and suppose it holds a lantern. Rule out what it lights, then follow the basics: numbers that fill up, dark cells with one way left. If two lanterns end up lighting each other, a number goes wrong, or a cell is left with no way to be lit, the cell is empty.

A lantern at row 7, column 8 (crossed) looks possible, but follow it a few steps and the board breaks. So it's crossed out. From Hard puzzle #1 — play it

It works the other way round too: suppose a cell has no lantern, and if that leaves some cell impossible to light, the lantern must be there. Good candidates are cells beside numbers and cells that light long lines. If a trial doesn't break within a few steps, undo it — our puzzles never need a long one.

Patterns worth knowing

A 4 always has a lantern on all four sides.
A 3 against the edge, or next to another wall, has only three open sides: all three are lanterns.
Diagonals of a 3. With four open sides, the cells diagonal to it are empty.
A 1 in a corner of walls with two open sides: one of the two is a lantern, so anything that would light both of them is empty.

Good habits

Numbers first. 0s, 4s and numbers against walls settle at once.
Watch the dark cells. The shading shows what's still unlit; a dark cell in a short corridor has very few ways to be lit.
Dot freely. Dots are free and undoable. Drag to dot a stretch at once.
Stuck? Hint names the technique and outlines the cells it settles.