Kakuro

• Every white square holds one digit from 1 to 9. Zero is never used.
• A run is an unbroken line of white squares, across or down. Each run has its own clue.
• The clue is printed in the shaded cell at the start of the run: above the diagonal for the run going right, below it for the run going down.
• The digits in a run must add up to the clue and must all be different from each other.
• A digit may repeat elsewhere in the grid — the no-repeat rule applies inside a run, not along the whole row.
• Every run here is between two and nine squares long, because nine distinct digits is all there is.

That is the whole game. There is no third rule, no shading, no outside knowledge. The difficulty comes from the fact that every square sits on two runs at once, so each digit has to satisfy an across clue and a down clue simultaneously.

Read it as a set of digits, not as arithmetic. "17 in two squares" cannot be 9 + 8 or 8 + 9 in any meaningful sense — it is the set {8, 9}, and the only question left is which square takes which. That reframing is the single biggest step a beginner can take, because it turns a subtraction problem into a lookup.

There are 120 clues a kakuro can print: every combination of run length from two to nine and a total that is actually reachable with that many different digits. Thirty-four of those 120 allow exactly one set of digits. On a finished grid here, roughly one clue in five is one of them, which is why a grid that looks impenetrable usually has three or four squares you can fill immediately.

The pattern is worth seeing rather than learning by rote: for any run length, the smallest total and the next one up are forced, and so are the largest and the one below it. The board's combination reference (press C, or find it in the three-dot menu) lists every set for whichever two clues cross the square you are sitting on, so nothing here needs to be memorised to play.

The ones in the middle. A total sitting at the centre of its range has the most ways of being made, and a four-square run totalling 20 is the worst clue in the game: twelve different sets of digits produce it, and so does a five-square run totalling 25. No other clue in kakuro reaches twelve. Those two tell you almost nothing on their own and have to be broken open from the runs crossing them.

Among the run lengths with room to have a middle — two through seven — four and five squares are the only ones whose worst total is a single number. The other four tie three ways: 9, 10 and 11 in two squares are all equally unhelpful, and so are 14, 15 and 16 in three. The table below therefore lists every total that ties rather than picking one of them to name. Eight and nine squares are a different matter entirely, because every total they can print allows exactly one set of digits.

Notice the two ends. A nine-square run can only total 45 and must contain every digit from 1 to 9 — there is nothing else nine different digits can be. An eight-square run is the same idea one digit short: each of its nine possible totals, 36 through 44, leaves out exactly one digit, so an eight-square run totalling 40 is 1–9 without the 5. Long runs look frightening and are the easiest clues on the grid.

No. Each puzzle here is built by filling a grid with digits, reading the clues off that fill, and then solving the result from scratch to check that no second answer exists. A fill that admits two answers is repaired and re-checked, so what reaches the screen is provably single-answer. If you are guessing, the deduction is there and you have not found it yet.

What that does not mean is that the puzzle falls to one trick. Looking at a square, listing what its across clue allows, listing what its down clue allows, and writing in the digit if only one survives — that technique alone never finishes a grid here. It settles a handful of squares and then stalls. The rest comes from working a whole run at a time: asking which digits a run cannot do without, and which of its squares can still take them.

Different, and harder to start. A sudoku hands you clues already written in the grid, so there is always somewhere to look. A kakuro starts completely empty, and the opening move has to come from a clue that constrains itself — a short run with an extreme total, or a long run with almost no freedom. Once the first few squares are in, kakuro tends to unravel faster than sudoku, because each new digit prunes two runs rather than three units that overlap heavily.

The arithmetic is also less of the puzzle than it looks. After the first few grids you stop adding up and start recognising clues — 16 in two, 7 in three, 45 in nine — the way a chess player recognises a position rather than calculating it.