Nonograms — also called Picross, Griddlers, or Paint by Numbers — hide a pixel picture inside a grid of number clues. Each row and column has a sequence of numbers indicating the lengths of filled runs in that line, in order. Fill the right cells and a picture emerges. The puzzle looks numerical but requires no arithmetic — just logic. And like all great logic puzzles, it has a systematic approach that prevents the frustrating dead ends most beginners encounter.
Start With the Most Constrained Lines
The cardinal rule of nonogram solving is to work the most constrained lines first. A constrained line is one where the clue numbers leave very little room to shift. The degree of constraint is easy to calculate: add up all the clue numbers for a line, add one for each gap between runs, and subtract from the line length. The result is the "slack" — how many positions each run can shift left or right.
A 10-cell row with a clue of [8] has a slack of 2: the run of 8 can start in position 1, 2, or 3. Cells 3 through 8 are guaranteed filled regardless of where the run starts. A 10-cell row with a clue of [1,1,1,1] has a slack of 6 — much higher — and yields no guaranteed cells at the start. Always work low-slack lines first and extract every certain cell before moving to higher-slack lines.
The Overlap Method
The guaranteed cells in a constrained run are found by the overlap method. For a run of length R in a line of length L, the leftmost possible position for the run starts at cell 1. The rightmost starts at cell L-R+1. The overlap — cells that are filled in both the leftmost and rightmost placements — is the cells from (L-R+1) to R, if that range is non-empty. These cells are certainly filled. Mark them immediately and move on.
When a line has multiple runs, apply the overlap method to each run separately, accounting for the minimum space required by the other runs. With full candidate notation (marking cells as definitely filled, definitely empty, or unknown), the overlap method typically fills a significant portion of every tightly constrained line on the first pass.
Using Filled and Empty Cells to Constrain Further
As you fill cells based on overlap logic, those filled cells constrain the possible positions of remaining runs. If a cell is confirmed filled in a line with multiple runs, it must belong to one of those runs — and knowing which run it belongs to pins that run's position, which in turn pins adjacent runs. Similarly, confirmed empty cells (from row constraints or completed runs) act as walls that further restrict where remaining runs can go.
After each wave of overlap-based fills, rescan every line for new constraints created by the confirmed cells. Nonogram solving is iterative — each confirmation creates new constraints that enable further confirmations. The puzzle rarely solves in a single pass. Work in waves: overlap sweep, update confirmed cells, rescan all lines, repeat.
When Logic Stalls: Hypothesis Testing
Well-constructed nonograms are always uniquely solvable without guessing. If you're stuck after thorough overlap sweeps and constraint propagation, you've missed something — return to the lowest-slack lines and re-examine them with fresh eyes. Check whether any recently confirmed cells have created new constraints you haven't applied. Check edge lines (first and last rows and columns) specifically — they're often more constrained than they appear because they can only extend in one direction.
If you're genuinely certain you've applied all constraints and the puzzle still has ambiguous cells, it may be a badly constructed puzzle with multiple solutions. In that case, a controlled hypothesis — try one value and propagate — is the appropriate next step. But for puzzles from reputable sources, this situation is rare. Trust the logic and keep scanning.