A Nonogram is a logic puzzle where you fill in squares on a grid to reveal a picture. Each row and column has a clue: a list of numbers giving the lengths of the runs of filled squares on that line, in order. Fill in the squares so that every row and column matches its clue.
The rules
- Each number is a run of filled squares, all next to each other on the line.
- The runs come in the order of the clue: left to right for a row, top to bottom for a column.
- Runs are separated by at least one empty square. There can be any number of empty squares before, between and after them.
That’s all. When every row and column matches its clue, the puzzle is solved, and the filled squares make the picture. You can also cross out squares you know are empty, as a reminder.
How a clue reads
Here’s a row of ten squares with the clue 3 1 2. Reading left to right: a run of 3, then a gap, then a single square, another gap, and a run of 2.
The clue doesn’t say where the runs start, or how wide the gaps are. That’s what the puzzle is about: the rows and columns cross, and each one narrows down the others.
A small example, solved
Here’s a 5×5 puzzle. The row clues are on the left and the column clues on top.
Step 1: start with the big clues
- Rows 2 and 3 need 5. The run fills all five squares, so both rows are filled.
- Columns 2, 3 and 4 need 4. A run of 4 in five squares can start in row 1 or row 2. Either way it covers rows 2, 3 and 4, so row 4 gets a filled square in each of those columns.
Step 2: cross out the finished lines
- Columns 1 and 5 need 2, and rows 2 and 3 already give them a run of 2. They’re finished, so their other three squares are crossed out.
- Row 4 needs 3, and has it. Its two end squares are already crossed out.
Step 3: row 1
Row 1 needs 1 1, and its end squares are crossed out, which leaves squares 2, 3 and 4. Two single squares with a gap between them fit in only one way: squares 2 and 4. Square 3 is crossed out.
Step 4: the point of the heart
- Column 3 needs 4, and its top square is crossed out. The 4 must fill rows 2 to 5.
- Columns 2 and 4 now have their 4, so their bottom squares are crossed out.
Row 5 now has its single square, and every row and column matches its clue.
Where to start in any puzzle
Look for clues that are big compared with their line. A quick way to measure that is the slack: the length of the line, minus what the clue needs (the runs plus one gap between each pair).
- A line of 10 with the clue 3 4 needs 3 + 1 + 4 = 8 squares, so the slack is 2.
- Any run longer than the slack has squares that are filled for certain: its length minus the slack. The 3 gives 1 square and the 4 gives 2.
You can see why by packing the runs as far left as they go, then as far right. The squares a run covers both times are filled whichever way the line turns out. We checked the slack rule against every possible clue for lines of 5 to 10 squares, and it holds for all 350.
On the game’s puzzles, about half the rows and columns give something away like this before you’ve filled a single square: 6 of the 10 lines on a 5×5, on average, about 10 of the 20 on a 10×10, and 13 of the 30 on a 15×15.
Common mix-ups
- Reading a clue as a total. A clue of 3 1 2 is three separate runs, not six squares in a row.
- Forgetting the gap. Two runs on the same line always have at least one empty square between them. A clue of 2 2 can’t be four filled squares in a row.
- Mixing up the order. The runs appear in the order of the clue: 1 3 is a single square and then a run of 3, never the other way round.