Sums with a no-repeat twist
Kakuro is often described as a crossword made of numbers. Instead of words you write digits, and instead of definitions you get sums. The arithmetic is never hard. The real puzzle comes from the second rule: no digit can repeat inside a run. That rule shrinks the number of ways to make each sum, and some sums can only be made one way at all.
Learn the fixed combinations
The fastest way to get better at Kakuro is to know which sums leave no choice. These are the smallest and largest totals for each run length.
- Two cells. 3 is always 1 and 2. 4 is always 1 and 3. 16 is always 7 and 9. 17 is always 8 and 9.
- Three cells. 6 is 1, 2 and 3. 7 is 1, 2 and 4. 23 is 6, 8 and 9. 24 is 7, 8 and 9.
- Four cells. 10 is 1, 2, 3 and 4. 11 is 1, 2, 3 and 5. 29 is 5, 7, 8 and 9. 30 is 6, 7, 8 and 9.
You do not need to memorise a full table. Notice the pattern instead: the lowest possible sum uses the smallest digits, and raising it by one swaps just the top digit up by one. The same works from the top end down.
Two more quick checks help with the in-between sums. A two-cell run with an even total can never use half that total, because it would need the same digit twice: a 10 in two cells cannot hold a 5. And a run's total caps its digits: in a two-cell run summing to 5, no cell can be larger than 4.
Crossing runs is where answers appear
Every white cell belongs to one across run and one down run. Selecting a cell lights both up, with their sums shown in amber, so make a habit of reading the pair together.
Here is a worked example on the 6×6 Easy board. Say a cell sits where an across run of two cells summing to 4 meets a down run of two cells summing to 3. The across run must be 1 and 3. The down run must be 1 and 2. The only digit they share is 1, so the crossing cell is 1. That immediately makes the other across cell 3 and the other down cell 2. Three cells from one glance.
When no digit is shared outright, narrow instead. If the across run can only use 1, 2 or 4 and the down run needs a digit of at least 5, the crossing cell has no option in common and you have made an error somewhere. If they overlap on just two digits, write them down mentally and look at the next crossing along.
Avoiding the usual traps
- Repeats hidden in long runs. On the 8×8 and 9×9 boards, runs grow longer and it is easy to place a digit that already appears further along. Read the whole run before you type.
- Guessing to fill space. This board has no candidate notes, so it is tempting to place a likely digit and see. Resist it. Work from the fixed combinations and crossings, and keep possible digits in your head or on paper.
- Forgetting the remaining total. When a run is partly filled, subtract what is there from the clue. A four-cell run of 20 with 9 and 8 placed needs 3 more across two cells, which can only be 1 and 2.
Getting better, one board size at a time
Easy is a 6×6 layout with a 5×5 area of white cells and short runs. Most decisions there come straight from the fixed combinations, so use it to drill them until you see "two cells, 17" and think "8 and 9" without counting.
Medium is 8×8, with longer runs that interact more. Practise the remaining-total habit there. Hard is 9×9, with longer chains where one placed digit feeds several crossings in turn.
If you get stuck, the hint fills one empty or clashing cell with its correct digit. Use it sparingly, and before continuing, check which run made that digit necessary. Clear empties the selected cell, and Reset wipes the board if you want to retry a puzzle from the start. For a related challenge that mixes sums with products and other operations, try Calcudoku at /games/calcudoku.
