Advanced Kakuro Techniques for Harder Grids

Once forced sums and crossing runs stop giving you cells, a bigger grid is not impossible, it is waiting for a sharper tool. These Kakuro techniques are for hard (10 x 10) and expert (12 x 12) grids. We show each one on the same grid, a hard LogikLand grid, and every number below was checked against its solution. New to Kakuro? Start with how to solve Kakuro.

Digits that are in every set

When a sum has several possible sets, look at what they have in common. A digit that appears in every set is in the run whatever happens.

In the grid above, column 3 (rows 6 to 10) has the clue 17 over five cells. So 1, 2 and 3 are in that column before you know where. The same reasoning tells you what is impossible: no 8 or 9 can be in that column, because both sets use only 1 to 7. Cross 8 and 9 off the five cells at once.

The technique becomes powerful when one digit is later placed. Suppose you find a 3 in the column, as we will in the block sum below. The remaining four cells must add up to 14 and still contain 1 and 2, so the other two digits make 11, and the only sets left are 1 2 4 7 and 1 2 5 6. The 3 has removed a third option.

Pairs that lock digits

Suppose two cells of the same run can each be only 4 or 7 after you crossed their lists. Those two cells will hold a 4 and a 7, in some order, so no other cell of the run can be 4 or 7. Cross them off the other cells. This is the same idea as a pair in Sudoku. In Kakuro it matters most in runs of 4 and 5 cells, where there are many more cells to clean. Look for it each time two cells of a run have the same two candidates written in them.

The block sum trick

This is the trick for the moment when no run gives you anything. Take a few across runs and look at the down runs that fit completely inside the same group of cells. The across clues add up to the same total as the down clues that cover the same cells. If the down runs cover all of the cells but one, that last cell is the difference.

First instance in the grid

In rows 5 and 6, on the left, take the across run with clue 16 (two cells in row 5) and the across run with clue 6 (three cells in row 6). Together they cover five cells. Two down runs lie entirely inside them: the clue 9 over the first column (rows 5 and 6) and the clue 10 over the second column (rows 5 and 6). They cover four of the five cells. Now 16 + 6 = 22 and 9 + 10 = 19, so the fifth cell, the third cell of row 6, is 22 − 19 = 3.

Second instance

On the right, in rows 5 and 6, the across clue 16 covers three cells in row 5 and the across clue 8 covers two cells in row 6. The down clues 14 and 3 sit inside the last two columns and cover four of those five cells. 16 + 8 = 24 and 14 + 3 = 17, so the leftover cell, the first cell of the clue-16 run, is 24 − 17 = 7.

The trick never needs a guess and never needs the candidates. It only needs you to see a block whose edges are clue cells. It works best where grey cells form a staircase.

The grid total as a check

In every Kakuro grid, the sum of all the across clues equals the sum of all the down clues, because both count every white cell once. In the grid above both totals are 339. If you copy a grid by hand, or if a printed grid seems to have no solution, adding the two totals is a quick way to find a clue that was read wrongly.

What the hard levels really need

The LogikLand generator keeps a Kakuro grid only if it can be solved by deduction. Its solver works run by run: for each run it keeps a digit in a cell only if some complete filling of the whole run (different digits, right sum, digits still allowed) uses it. That includes digits that are in every set and pairs, done systematically. The block sum trick is not part of that solver, so you never need it, but when you see one it is a shortcut. Hard grids have runs of up to 5 cells on a 10 x 10 area, expert grids the same run length on a 12 x 12 area.

Practising

Try the techniques in this order: forced sums, crossings, always-present digits, pairs, block sums. Our suggestion, not a rule: print a hard grid with a wide margin so you have room for candidates. To make hard and expert grids for a book, create a free account and open the Kakuro assistant. In 6 x 9 books with hard or expert grids, the assistant allows one grid per page, because the clue numbers become small. For easier practice, see printable easy Kakuro puzzles.