A crossword made of sums

Kakuro

Kakuro looks like a crossword built from numbers, and it is sometimes called cross sums. The black cells hold the clues, the white cells take digits, and every answer has to add up. Choose a level from the tabs above the board: there are 20 puzzles at each of easy, medium, hard and expert.

Kakuro
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The arithmetic never goes beyond adding single digits. What makes a kakuro hard is knowing which digits can make a total, and spotting where an across run and a down run have to agree.

The rules

  1. Write a digit from 1 to 9 in every white cell.
  2. A run is an unbroken line of white cells, across or down. The clue in the black cell to the left of an across run, or above a down run, is the total of that run.
  3. No digit may appear twice in the same run. The same digit can appear again in the same row or column if it sits in a different run.
  4. Each white cell belongs to one across run and one down run, so it must satisfy both clues.

Your first moves

Some totals can only be made one way, and they are the way in. In two cells, 3 is 1 and 2, 4 is 1 and 3, 16 is 7 and 9, 17 is 8 and 9. In three cells, 6 is 1, 2 and 3, 7 is 1, 2 and 4, 23 is 6, 8 and 9, and 24 is 7, 8 and 9. In four cells, 10 is 1 to 4 and 30 is 6 to 9.

Now look where two of those runs cross. If a 16 in two cells crosses a 17 in two cells, the shared square must be 9, the only digit both runs allow. A 3 crossing a 4 gives you a 1 the same way. When a run has only one empty cell left, that cell is simply the clue minus the digits already placed.

How the levels and sizes differ

Easy puzzles are played on an 8×8 board, counting the clue border, and need nothing beyond those combination and crossing moves. Medium and hard both use a 10×10 board; expert uses 12×12, with longer runs to keep track of.

Medium adds combination elimination. A 12 in three cells could be 1, 2, 9 or 1, 3, 8 or several other sets, but if none of the three cells can hold a 9, every set containing 9 is gone. It also brings hidden singles: a digit every surviving set needs, which only one cell in the run can take.

Hard puzzles add pairs and triples inside a run, and remainder reasoning on the last two cells. If two cells must make 10, neither can be 5, because the other would have to be 5 as well. Expert asks for full run analysis: listing every way the open cells of a run can reach its total with different digits, then keeping only the digits that occur in at least one of them. It is slow on paper and very satisfying when it breaks a long run open.

Rules, cheat sheet and printables

The kakuro rules page covers the details, and how to solve kakuro goes through a complete puzzle move by move. Keep the kakuro cheat sheet of combinations nearby until the common totals stick.

For paper solving, the printable kakuro packs put the answers on the last pages, and large print kakuro uses bigger cells and clue numbers that are easier to read.