How to Play Kakuro: Rules, Sums and Beginner Solving Techniques

Clue squares and answer squares

A Kakuro grid has two kinds of square. White squares are where you write your answers. Clue squares are shaded and split by a diagonal line; the numbers inside them are the clues. Fully shaded squares with no number are simply blank.

Across and down runs

A line of white squares running unbroken across or down is called a run (or an entry). Every run starts right next to a clue square, and the clue gives its total:

• The number above the diagonal, in the top-right half, is the across clue. It gives the total of the white squares to its right, up to the next shaded square or the edge.
• The number below the diagonal, in the bottom-left half, is the down clue. It gives the total of the white squares beneath it.

An easy way to remember: each number sits on the side of the line that faces its run. The top-right number looks across to the right; the bottom-left number looks down.

Every white square belongs to exactly two runs: one across and one down.

The rules

1. Write one digit from 1 to 9 in every white square. There is no zero.
2. The digits in each run must add up to its clue.
3. No digit may repeat within a run.

The third rule applies to each run, not to the whole row or column. In the example above, the top row reads 1, 3 and then 1, 3 again. That’s fine: a shaded square separates them, so they are two different runs. Within either run, every digit is different.

As with any well-made puzzle, there is exactly one solution and logic alone will find it.

Combinations with only one answer

Because digits can’t repeat and must be between 1 and 9, some totals can only be made one way. Learn these and you’ll always have a place to start:

Why does 4 in two squares have to be 1 and 3? The other way to make 4 is 2 + 2, and that repeats a digit. And 17 in two squares can only be 8 + 9, because 9 is the biggest digit allowed.

Which digits, not which order

A combination tells you which digits go in a run, not where. A two-square run totalling 3 could read 1, 2 or 2, 1. To decide the order, look at the run that crosses each square.

Most totals have several combinations. Ten in two squares, for example, could be 1 + 9, 2 + 8, 3 + 7 and 4 + 6. (5 + 5 isn’t allowed.) For these, you need help from the crossing clues.

Where clues cross

The square where two runs meet must satisfy both. That’s the heart of Kakuro: put the two lists of possible digits side by side and keep only what they share.

• 16 in two squares can only be 7 + 9.
• 17 in two squares can only be 8 + 9.
• The shared square must be in both lists. The only digit in both is 9.

With a 9 in the corner, the rest follows: the across 16 needs a 7 beside it, and the down 17 needs an 8 beneath it.

Finishing a run by subtraction

When all but one square of a run is filled, the last square is simply the clue minus what’s already there. A three-square run totalling 15 that already holds 6 and 2 needs 15 − 6 − 2 = 7.

Subtraction is also a useful check. If a run of 10 already holds 3 and 4, the last square would have to be 3. That repeats a digit already in the run, so one of the earlier digits must be wrong: time to retrace your steps.

Smallest and largest sums

For any length of run there is a smallest and a largest possible total. The smallest uses the lowest digits (1 + 2 + 3…), the largest the highest (9 + 8 + 7…). The totals right next to those limits can only be made one way, which makes them another reliable starting point:

When a run has more than one combination, write the possible digits as small pencil marks, then use the crossing runs to cross them off. A digit that appears in every combination of a run is certain to be in it somewhere, even before you know exactly where.

A complete worked puzzle

Here is a small puzzle with two across clues and three down clues. Try it yourself first, or step through it below. Each step shows which run it uses and why the digits are forced.

Mistakes and tips

• Reading a clue in the wrong direction

  Top-right half reads across, bottom-left half reads down. Check which side of the diagonal each number is on.

• Repeating a digit within a run

  8 in two squares can’t be 4 + 4. Every digit in a run must be different.

• Applying the no-repeat rule to the whole row

  A digit can appear again in the same row or column if a shaded square separates the runs.

• Fixing the order too early

  Knowing a run contains 1 and 2 doesn’t tell you which is first. Wait for a crossing clue to decide.

Tips

• Start with the totals that have only one combination, especially short runs.
• Work where two such runs cross: the shared square is often decided immediately.
• Use pencil marks for the possible digits, and cross them off as crossing runs rule them out.
• Check each finished run by adding it up. It only takes a moment and catches slips early.

A puzzle to try

Ready for one on your own? This puzzle is new, and twice the size of the worked one. Everything you need is in this guide: combinations with only one answer, crossing runs and subtraction.

Quick rules to print

A one-page reminder to keep beside your puzzle book.