How to use the kakuro and killer sudoku combination finder
- Enter the number of cells in the cage or run, from 1 to 9.
- Enter the sum the clue gives.
- If you already know digits that must be in the cage, list them. If you have ruled out digits, list those to leave out.
- Read the combinations, the digits shared by all of them and the digits that cannot be used.
Formula
Choose n different digits from 1 to 9 with total equal to the clue
Least sum for n cells = n (n + 1) / 2; greatest = n (19 - n) / 2
Keep the sets that contain the digits you know are in
Drop the sets that use a digit you have ruled out
The number of different digit sets for n cells and every possible sum adds up to the number of ways to choose n digits from 9, for example 84 for three cells.
Cages of two, three and four cells
Three cells adding to 10 have 4 sets: 1 + 2 + 7, 1 + 3 + 6, 1 + 4 + 5 and 2 + 3 + 5. Two cells adding to 17 have only 8 + 9, and four cells adding to 10 have only 1 + 2 + 3 + 4. Five cells adding to 15 have just 1 + 2 + 3 + 4 + 5.
Three cells adding to 15 give 8 sets, and if one cell must be a 1, only 1 + 5 + 9 and 1 + 6 + 8 are left, so the 1 is in every set. Leaving out the digit 9 leaves 6 of the 8 sets.
Using combinations in a puzzle
In both Kakuro and Killer Sudoku a group of cells must hold different digits that add to a number. The list of possible sets is the first thing to check. A digit that appears in every set can be written in as a candidate for the cage, and a digit that appears in none can be crossed off. When a clue leaves only one set, you know exactly which digits go in, and only their order is open.
More puzzle tools are the sudoku solver, the sudoku generator, the 24 game solver and the combinations and permutations calculator.
Extremes to remember
Some clues have a single answer and are worth learning: 3 for two cells (1 and 2), 4 for two cells (1 and 3), 16 and 17 for two cells, 6 for three cells (1, 2 and 3), 23 and 24 for three cells, 10 for four cells, and 29 and 30 for four cells. The page confirms them and shows every other case.
Assumptions and limits
- The cells hold different digits from 1 to 9, as in Kakuro runs and Killer Sudoku cages.
- The sum must be reachable: for example, two cells add to between 3 and 17.
- Digits to include or leave out are entered as numbers from 1 to 9, separated by commas or spaces.
- The order of digits in the cells is not decided; the page lists sets.
Frequently asked questions
What are the combinations for three cells that add to 10?
The sets are 1, 2, 7, then 1, 3, 6, then 1, 4, 5 and 2, 3, 5. That is four sets in all.
How do I find a Kakuro combination?
List the sets of different digits with the right size and sum. The page does this and can narrow the list using digits you already know.
Is a Killer Sudoku cage the same as a Kakuro run?
For this purpose, yes. Both need different digits from 1 to 9 that add to the clue, so the same list of sets applies.
Why are some sums impossible?
The smallest sum for n different digits is 1 + 2 + ... + n and the largest uses the n biggest digits. A clue outside that range cannot be made.