Kakuro

How does Kakuro work?

Kakuro is a crossword made of numbers. Every white square takes a digit from 1 to 9. Each run of white squares, across or down, must add up to the clue at its start, and no digit may appear twice in the same run. In a split clue square, the number above the line is for the run to its right, and the one below is for the run beneath.

Choose a 4 × 4 white area for a few minutes, or 6 × 6 for a longer puzzle. Select a square, then tap a digit or type it. A run with a repeated digit, or one whose digits already go past its clue, is outlined in red as you go; that comes from the clues alone, not from the answer.

How is your Kakuro time calculated?

Your result is the time from pressing Start to the moment the last square is right. Next to it are the times you pressed Check, which marks any digit that disagrees with the answer, and Hint, which fills in one square (or fixes a wrong one). Both are counted, never hidden. There is no rating: no published typical time exists for puzzles built like these, and a band without data would be invented.

Times vary with the puzzle more than with anything else, since some grids have one stubborn corner. Typing digits on a keyboard is a little quicker than tapping them on a phone, and the clock keeps running while you think, so a break counts too.

How do you solve Kakuro?

Start with the runs that can only be filled one way. A two-square run adding to 3 must be 1 and 2, and one adding to 17 must be 8 and 9. Then work where an across run and a down run cross: the square has to take a digit both runs allow. Most runs allow several sets; two squares making 10 can be 1 + 9, 2 + 8, 3 + 7, 4 + 6.

These are the short runs with only one set of digits, worked out by the site's code. They are often called a Kakuro cheat sheet:

Longer runs have them too: in all, 34 combinations of length and clue have exactly one set, among them every run of eight squares and the nine-square run adding to 45. The order of the digits inside a run is what the crossing runs decide.

• Use the biggest and smallest totals. Three squares can add to at most 24 (7 + 8 + 9) and at least 6 (1 + 2 + 3), so a clue near either end leaves little choice.
• Subtract what is known. Once two squares of a four-square run are filled, the other two must make the rest of the clue, with digits not used yet.
• Use Check sparingly. It tells you which digits are wrong, which is a big clue; the count is on your result.

Does every Kakuro puzzle have exactly one answer?

Yes. The 60 puzzles here were made ahead of time, and each one was kept only if a solver that never guesses could work out every square from the clues, using the same run-by-run reasoning as above. The site's tests solve every stored puzzle again and count its answers by trying every possibility, so each has exactly one. Half the time a puzzle is shown with its rows and columns swapped, which keeps every sum true.

Is Kakuro harder than Sudoku?

It is different rather than harder. Sudoku asks only where each digit goes; Kakuro adds sums, so each step is a small piece of arithmetic as well as logic. The size of the grid matters more than the name: the 4 × 4 puzzles here are quick, while newspaper Kakuro often runs to grids of 10 × 10 or more.

What does Kakuro mean?

The name is short for the Japanese kasan kurosu, "addition cross". In English the puzzle is also called Cross Sums, the name Dell Magazines used; Canadian Jacob E. Funk, who worked for Dell, coined it in 1966. In Japan it is one of the best-known puzzles of the publisher Nikoli, second only to Sudoku.

For another puzzle of sums, try Cross Math; for one where every line must add up to the same total, the Magic Square.