Nonogram solving techniques

If you know the rules and can solve small grids, these are the techniques that carry you through 12×12 and 15×15 puzzles. Each one is a way of squeezing certainty out of a line before you know everything about it.

Overlap (the workhorse)

Push a run as far left as it can go, then as far right, and fill every cell covered in both extremes. The bigger the clue relative to the free space, the more it pays. In a 15-wide line, a 10 fixes five cells immediately; a 4 4 4 fills nothing on its own — until crossing lines shrink its slack.

Slack is the number a solver watches: line width minus (sum of clues + minimum gaps). A slack of 0 solves the whole line instantly. Overlap works on any run longer than its slack.

Edge logic

A filled cell touching a wall — or touching a cell you’ve marked × — anchors the nearest run. If the first clue is 3 and the first cell of the line is filled, cells 2 and 3 are filled too, and cell 4 is ×. Edges are why finished corners make the whole neighbourhood fall quickly.

Glue

A filled cell near a wall (but not touching it) still constrains the run it belongs to. First clue 3, and cell 2 is filled? The run can start at cell 1 or 2, so cell 3 is certainly filled — the known cell “glues” part of the run in place even though you don’t know its exact position.

Gap elimination

Mark the cells no run can reach. If a line’s clue is 4 and a stretch between two × marks is only three wide, every cell in that stretch is ×. Checking each gap against the smallest clue that could live there is often what unsticks a stalled line.

Punctuate finished runs

The moment a run reaches its full length, put an × at both ends. It feels minor; it is not — each × is a wall for the crossing column, and cascades follow. Most “stuck” boards have several finished-but-unpunctuated runs waiting to pay out.

Count from both directions

When a line has many clues (2 1 1 3 in 12 cells), number the runs and work out each run’s leftmost and rightmost possible position. Any cell where the same run appears in both numberings belongs to that run. This is overlap generalised to multi-run lines, and it is the main tool on dense 15×15 rows.

Sweep, don’t stare

Every determined cell adds information to one row and one column. Strong solvers don’t exhaust a line — they take the cheap deductions and move on, returning after the crossing lines have changed. If a line yields nothing in a few seconds, leave it; it will be easier when you come back.

Every puzzle here is verified solvable with exactly these techniques — no contradiction chains, no bifurcation, no guessing. If you’re stuck, some line on the board has a deduction waiting. Practice on the medal boards or today’s daily puzzle.