Kakuro

Learn the magic blocks

Kakuro looks like arithmetic and plays like logic, and the bridge between the two is a short list of sums that can only be written one way. A two-cell run adding to 3 has to be 1 and 2. A two-cell run of 17 can only be 8 and 9. Three cells making 6? That's 1, 2 and 3 in some order. Three cells making 24 is 7, 8 and 9. Solvers call these forced combinations magic blocks, and they're where you start, because they hand you a tiny set of digits before you've deduced anything else.

The rule that makes them work is the one people skip on their first puzzle: no digit repeats inside a single run. Every white cell belongs to one across run and one down run, so a digit that fits the across clue also has to fit the down clue crossing it. That overlap is the whole game.

Working an intersection

Say a two-cell across run totals 4 (that's 1 and 3, never 2 and 2, since digits can't repeat), and one of those cells sits on a two-cell down run of 16, which must be 7 and 9. The shared cell has to be a value both runs allow. The across side offers 1 or 3, the down side offers 7 or 9, and they share nothing, so either you've misread a clue or you've narrowed a neighbour hard. Chasing those clashes is faster than guessing.

Now run the same logic in a friendlier spot: an across pair totalling 4 (so 1 and 3) crossing a down pair totalling 3, which is 1 and 2. The only digit both runs accept is 1, so the shared cell settles instantly, the across partner becomes 3, the down partner becomes 2, and three cells fall out of one observation. That's the rhythm of the whole game: every solved cell tightens its run, every tightened run leans on the next crossing, and a grid that looked blank ten minutes ago starts collapsing in chains.

• Pencil in the candidate digits for the tightest runs first and let the crossings trim them.
• There are no zeros anywhere: every white cell is a 1 through 9.
• A long run with a low total, or a short run with a high one, is the most constrained. Start there.

The 45 trick and the blocks worth memorising

The digits 1 through 9 add to 45, and that one number solves cells nothing else reaches. A nine-cell run always totals exactly 45. An eight-cell run totals 45 minus its single missing digit: a clue of 38 tells you the absent digit is 7 before you've filled in a thing. Mid-sized runs reward the same subtraction habit: a seven-cell run's clue fixes which two digits are missing, and the crossings usually allow only one way to place them. Past the 45 rule, a short list of forced combinations carries most grids:

• Two cells: 3 is 1+2, 4 is 1+3, 16 is 7+9, 17 is 8+9.
• Three cells: 6 is 1+2+3, 7 is 1+2+4, 23 is 6+8+9, 24 is 7+8+9.
• Four cells: 10 is 1+2+3+4, 11 is 1+2+3+5, 29 is 5+7+8+9, 30 is 6+7+8+9.

Notice the mirror: every low block has a high twin, because swapping each digit for ten-minus-itself turns a 3 into a 17 and a 6 into a 24. Learn the low half and the high half comes free.

Mistakes that stall a grid

Guessing is the big one. These grids have a single solution reachable by logic alone, and a guess that happens to satisfy its own run can sit there looking innocent while it quietly poisons three others. By the time the contradiction surfaces, you won't remember which cell to blame. When you're stuck, re-derive candidates for the tightest runs instead of gambling. The quieter mistakes: forgetting the no-repeat rule inside long runs, where a five-cell run swallows a duplicate without the eye catching it; mixing up the 16 and 17 blocks, which plants an 8 where a 7 belongs and detonates minutes later; and skipping the cleanup after each placement. Every digit you write eliminates that digit from every other cell of both its runs. Do that small bookkeeping every single time, and grids that felt impossible start feeling merely stubborn.

Cross sums by another name

The puzzle began in the United States as Cross Sums, a Dell puzzle-magazine staple from the mid-1960s, long before it became a Japanese fixture. Nikoli imported it, and the company's founder Maki Kaji (the same man who later coined the name Sudoku) contracted the Japanese phrase for addition-cross into Kakuro. At home it spent years among Nikoli's most popular puzzles, and when Sudoku swept the world in 2005, British newspapers reached for Kakuro as the follow-up act. Same idea underneath (fill the grid so every run hits its clue), but the no-repeat rule gives it a flavour Sudoku doesn't have: something closer to a crossword where the dictionary happens to be arithmetic.