Kakuro — free online cross sums puzzle

Kakuro looks like a crossword and behaves like arithmetic. The white squares come in runs — a horizontal strip, or a vertical one — and each run has a clue printed in the shaded square at its head: the number above the diagonal is the total for the run going right, the number below it is the total for the run going down. Fill every white square with a digit from 1 to 9 so that each run hits its total, and no digit appears twice in the same run. A digit can repeat in a row, as long as the two are in different runs.

That no-repeat rule is what turns addition into deduction. Two squares totalling 16 can only be 7 and 9; three totalling 6 can only be 1, 2 and 3. Every grid here is generated fresh and then checked to have exactly one solution before you see it, so it is always finishable by reasoning and never by guessing. Free, no download, and a part-finished grid is saved if you close the tab.

How to play

• Tap a white square, then tap a digit on the pad. Tap ⌫ Erase to clear it.
• Each run of white squares must add up to the clue at its head — above the diagonal for across, below it for down.
• No digit twice in the same run. Only 1 to 9 are used; there is no zero.
• Turn on Notes to pencil in the candidates for a square.
• A digit that repeats in its run, or pushes a run past its total, turns red as you place it.
• Pick Small, Medium or Large on the menu. Each size keeps its own list of best times.

The combinations worth memorising

Kakuro rewards a very short list of facts. Because a run cannot repeat a digit, some totals have only one possible set. For two squares: 3 is 1+2, 4 is 1+3, 16 is 7+9 and 17 is 8+9. For three squares: 6 is 1+2+3, 7 is 1+2+4, 23 is 6+8+9 and 24 is 7+8+9. For four: 10 is 1+2+3+4 and 30 is 6+7+8+9. You do not know the order yet, but you know the contents, and a crossing run that shares one of those squares usually gives up the order immediately.

The second habit is to look where two of those locked runs cross. If an across run must contain {7,9} and the down run through one of its squares must contain {1,2,3}, that square is in both sets — and since they share nothing, that layout is impossible, so you have learnt something about the rest of the grid. When they do share exactly one digit, that square is solved outright. Working the crossings is the whole game; adding up is only how you get there.

Kakuro is the older of the two ideas. It ran in American puzzle magazines under the name "cross sums" for years before Sudoku became a fixture, and the family resemblance is not a coincidence: both are built on the same constraint, that a group of cells has to hold distinct digits. The difference is where the constraint bites. Sudoku puts it on rows, columns and boxes of nine; Kakuro puts it on short runs and then tells you what they add up to. If you like the arithmetic, Killer Sudoku applies it to a Sudoku grid, and plain Sudoku is the version with no addition at all.

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• Minesweeper — The desk-drawer classic: 9×9 grid, 10 hidden mines. Read the numbers, flag the bombs, clear the board — one wrong tap and boom.
• Sudoku — Classic 9×9 logic puzzle with pencil notes and error checking. Three difficulties, every puzzle guaranteed one unique solution.
• 15 Puzzle — The classic sliding tile puzzle. Slide 15 numbered tiles back into order — fewest moves and fastest time both saved.

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