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Picture by numbers

Fill the squares the row and column numbers describe — each number a run of filled squares, in order. 5×5 up to 10×10, one picture each.

  • Logic
  • Untimed
  • 10 levels

How it plays

Beside each row and above each column is its clue: the lengths of the runs of filled squares in that line, in order, with at least one empty square between runs. Fill a square ■ or mark it ·.

Overlap does most of it: a run of 4 in a line of 5 covers the middle three squares wherever it sits. Mark what is certain, then use it in the crossing lines, and go round again.

Levels grow the picture, and the harder ones may need a guess followed to a contradiction. Every puzzle has exactly one solution, checked by a line solver with backtracking.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This practises systematic case analysis on lines — what holds in every placement. It is practice at these puzzles, not a measured gain in reasoning elsewhere.

Learn

How to get better at picture by numbers: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Mark every line whose clue is 0 as empty, all the way across.
  2. For each line, work out its slack: the line’s length minus the clue’s total and the one-square gaps between runs. Start with the lines that have the least.
  3. Fill the overlap of every run longer than the slack, then turn to the crossing lines: each new square you mark changes what they can do.
  4. Go round rows, then columns, then rows again, until a full pass changes nothing. When a line’s filled squares match its clue, mark the rest of it empty.
  5. If you stall on a puzzle that may need a guess, pick one square, assume it filled, and follow the consequences until the picture breaks or finishes.

Techniques and tricks

  • Slack and overlap

    Add the runs and one gap between each; take that from the line’s length to get the slack. Each run longer than the slack has its length minus the slack filled for certain, in the middle of its range.

    e.g. Line of 10, clue 3 4: 3 + 1 + 4 = 8, slack 2. The 3 fixes square 3; the 4 fixes squares 7 and 8.

  • Zero slack fills the line

    When the runs and their gaps add up to the line’s length exactly, the line has one placement. Fill it in at once, gaps and all.

    e.g. Line of 7, clue 2 1 2: 2 + 1 + 1 + 1 + 2 = 7, so ■■·■·■■.

  • Glue to the edge

    A filled square close to an edge belongs to the first run on that side. The run must cover it, so it can reach only so far from the edge; squares past that are empty.

    e.g. Line of 10, clue 4, square 2 filled: the run covers squares 2–4 whatever happens, and squares 6–10 are empty.

  • Squeeze past an empty square

    A marked empty square splits a line. If a run cannot fit in the gap before it, that gap is empty and the run’s range starts after it — then redo the overlap in the shorter range.

    e.g. Line of 10, clue 4, square 3 empty: the 4 cannot fit in squares 1–2, so they are empty, and the 4 lives in squares 4–10, which fixes square 7.

  • Count to close a line

    Add up a line’s clue. Once that many squares are filled and they form the right runs in the right order, every other square in it is empty.

  • Guess, then follow to a contradiction

    On the puzzles that may need it, pick a square in a tight line and assume it is filled. Keep line solving; if some line can no longer fit its clue, the square was empty. Remember which squares came from the guess so you can undo them.

As the levels climb

  • The ladder is a fixed list of up to ten rungs, cut short by “Largest picture”: two 5×5s, then 6×6 and 7×7, each solvable line by line with no guess. Build the slack-and-overlap habit here.
  • From the second 7×7 on, some rungs are marked “may need a guess”. Line solving still does most of the work; only reach for a guess once a full pass of rows and columns finds nothing.
  • The pictures grow to 8×8, 9×9 and 10×10, and the clues get longer. Lines with three or four runs have little overlap, so lean on the crossing lines and on squares near the edges.
  • Where the clues alone would not settle one picture, a few squares are shown already filled or marked. Treat them as clues: glue and squeeze from them first.

Traps to avoid

  • Forgetting the gap: two runs need at least one empty square between them. Leave it out of the slack sum and every overlap you work out is wrong.
  • Reading the clue out of order: 1 3 means the 1 comes before the 3, reading left to right or top to bottom. Placing the 3 first fits the count but not the clue.
  • Taking overlap from one run alone: a run’s range is limited by the runs on either side of it. Slide the whole clue to the left, then the whole clue to the right, and compare run by run.
  • Matching the count but not the runs: clue 1 2 means three filled squares, but as a single, a gap, then a pair. A solid block of three fits the count and breaks the clue.
  • Guessing too early: a guess on a line-solvable puzzle is wasted work, and on any puzzle it is easy to lose track of which squares came from it.

How to practise

  • Set “Largest picture” to 6 or 7 and “Solving” to “Always line by line” until slack and overlap come without counting on your fingers.
  • Then put “Solving” back to “Guesses at the higher levels” and let the larger pictures in, one size at a time.
  • Keep “Time pressure” off while learning. With it on, the clock gives a minute and a half plus five seconds for every open square.
  • After each puzzle, read the first step in the working and check you would have found the same tightest line.

Levels

It is untimed unless you turn the clock on. There are 10 levels with the default settings, and the counter stops at the real top. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

More logic

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