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How to Solve Kakurasu: 6 Techniques That Work

To solve Kakurasu, work one row or column at a time: list the ways its target can be made, then shade the squares every way uses and cross out the squares none of them use. Each square you settle narrows down the line crossing it, and the puzzle unravels from there. Six techniques cover all of it, and each one is shown below before we use them together on a full 5×5 puzzle.

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New to Kakurasu? Read the Kakurasu rules first. This page assumes you know that a shaded square counts its column number towards its row, and its row number towards its column.

Think in combinations

Every technique here comes down to one question: which sets of squares add up to this line’s target? In a 5×5 grid there are never more than three. Here they all are:

Target Combinations
1 1
2 2
3 1 + 2, 3
4 1 + 3, 4
5 2 + 3, 1 + 4, 5
6 1 + 2 + 3, 2 + 4, 1 + 5
7 1 + 2 + 4, 3 + 4, 2 + 5
8 1 + 3 + 4, 1 + 2 + 5, 3 + 5
9 2 + 3 + 4, 1 + 3 + 5, 4 + 5
10 1 + 2 + 3 + 4, 2 + 3 + 5, 1 + 4 + 5
11 1 + 2 + 3 + 5, 2 + 4 + 5
12 1 + 2 + 4 + 5, 3 + 4 + 5
13 1 + 3 + 4 + 5
14 2 + 3 + 4 + 5
15 1 + 2 + 3 + 4 + 5

Notice the mirror: the combinations for 11 are the squares left out by the combinations for 4, and so on down the table. That’s the fourth technique below.

1. Start with the targets that have one way

At every grid size, five targets can only be made one way: 1, 2 and the three largest. In a 5×5 grid that’s 1, 2, 13, 14 and 15. Fill these lines in completely before anything else; the rules page has the table for every grid size. More on one-way targets.

2. Too big to fit

A square worth more than the target can’t be in the line. In a 7-square row with target 5, squares 6 and 7 are crossed out straight away.

Too big to fit: a 7-square row with target 5 A row of seven squares with target 5. Squares 6 and 7 are crossed out, because each is worth more than 5 on its own. 1 2 3 4 5 6 7 5
Target 5: squares 6 and 7 are each worth more than 5, so both are crossed out.

This works on what’s left of a target, too. If a row needs 7 and already has square 5 shaded, only 2 remains, so every unknown square worth more than 2 is out.

More on too big to fit.

3. Can’t do without it

Ask of each big square: could the rest of the line reach the target without it? If not, it must be shaded. In a 5-square row with target 12, the other squares make at most 1 + 2 + 3 + 4 = 10 without square 5, and 1 + 2 + 3 + 5 = 11 without square 4. Both fall short, so squares 4 and 5 are shaded. Without square 3 the rest make 12, which is enough, so square 3 stays open.

Can't do without it: a 5-square row with target 12 A row of five squares with target 12. Squares 4 and 5 are shaded: without square 5 the rest make at most 10, and without square 4 at most 11. 1 2 3 4 5 12
Target 12: without square 5 the most you can make is 10, and without square 4 it’s 11. Both squares must be shaded.

This and the next technique are the same rule put two ways: they always settle exactly the same squares. Use whichever comes more easily. Both are covered in counting the empty squares.

4. Count the empty squares

A line’s shaded and empty squares always add up to the largest target, so a big target is a small target for the empty squares. In a 7-square row the largest target is 28. A target of 25 means the empty squares add up to 3, which is either square 3 or squares 1 and 2. Either way, squares 4 to 7 are shaded.

Count the empty squares: a 7-square row with target 25 A row of seven squares with target 25. The empty squares must add up to 28 minus 25, which is 3, so they are square 3 or squares 1 and 2. Squares 4 to 7 are shaded. 1 2 3 4 5 6 7 25
Target 25 of 28: the empty squares add up to 3, so they’re among squares 1 to 3, and squares 4 to 7 are shaded.

Use this whenever a target is more than half the largest. It turns every big target into a small one. More on counting the empty squares.

5. Find what every combination shares

List the combinations and compare them. A square used by every combination is shaded; a square used by none of them is crossed out.

A 5-square row with target 11 has two combinations, 1 + 2 + 3 + 5 and 2 + 4 + 5. Both use squares 2 and 5.

What every combination shares: a 5-square row with target 11 A row of five squares with target 11. Its two combinations, 1 + 2 + 3 + 5 and 2 + 4 + 5, both use squares 2 and 5, so those are shaded. 1 2 3 4 5 11
Target 11: both combinations use squares 2 and 5, so both are shaded.

A 5-square row with target 4 can be 4 or 1 + 3. Neither uses square 2, and neither uses square 5.

What no combination uses: a 5-square row with target 4 A row of five squares with target 4. Its combinations are 4 and 1 + 3. Neither uses square 2 or square 5, so both are crossed out. 1 2 3 4 5 4
Target 4: no combination uses square 2 or square 5, so both are crossed out.

Techniques 2 to 4 are quick special cases of this one. When they don’t apply, this always does. More on shared squares.

6. Let crossing lines cut the combinations

In a 5×5 grid, targets from 5 to 10 tell you nothing on an empty line: their combinations between them use every square and leave every square out. They open up once a crossing line settles a square. A shaded square rules out every combination that leaves it out; a crossed-out square rules out every combination that uses it.

This is the engine of every solve. Each time you shade or cross out a square, look at the line running the other way through it. The worked example below shows it happening: a column with three possible combinations drops to one because of a single crossed-out square. More on crossing lines.

A full 5×5 puzzle, solved step by step

Here’s a puzzle that leans on the middle targets, so most of it is crossing-line work.

A 5 by 5 Kakurasu puzzle An empty 5 by 5 grid. Row targets from top to bottom: 10, 6, 3, 10, 7. Column targets from left to right: 10, 7, 9, 10, 3. 1 2 3 4 5 1 2 3 4 5 10 6 3 10 7 10 7 9 10 3
Rows need 10, 6, 3, 10 and 7. Columns need 10, 7, 9, 10 and 3.

Step 1: the small targets

Only two lines have targets outside 5 to 10: row 3 and column 5, both 3. Row 3’s combinations are 1 + 2 and 3, so squares 4 and 5 are too big (technique 2). That crosses out row 3 of column 5, which leaves column 5 only one combination: 1 + 2. Shade rows 1 and 2 of column 5 and cross out rows 4 and 5.

Step 1: row 3 and column 5 Row 3 (target 3) crosses out squares 4 and 5. Column 5 (target 3) then has row 3 crossed out, so it must be rows 1 and 2: those are shaded and rows 4 and 5 crossed out. 1 2 3 4 5 1 2 3 4 5 10 6 3 10 7 10 7 9 10 3
Row 3 rules out squares 4 and 5. With row 3 gone, column 5 must be 1 + 2.

Step 2: column 4

Column 4 needs 10, which could be 1 + 2 + 3 + 4, 2 + 3 + 5 or 1 + 4 + 5. Row 3 is crossed out, and the first two both use it. Only 1 + 4 + 5 is left (technique 6): shade rows 1, 4 and 5 and cross out row 2.

Step 2: column 4 Column 4 (target 10) could be 1 + 2 + 3 + 4, 2 + 3 + 5 or 1 + 4 + 5. Row 3 is crossed out, which leaves only 1 + 4 + 5: rows 1, 4 and 5 are shaded and row 2 crossed out. 1 2 3 4 5 1 2 3 4 5 10 6 3 10 7 10 7 9 10 3
A single crossed-out square takes column 4 from three combinations to one.

Step 3: rows 1, 2 and 4

Each of these now has one combination left:

  • Row 1 needs 10 and has 4 + 5 = 9. It needs 1 more, so square 1 is shaded and squares 2 and 3 are out.
  • Row 2 needs 6 and has square 5. It needs 1 more, and square 4 is already out, so square 1 is shaded and squares 2 and 3 are out.
  • Row 4 needs 10 and has square 4, with square 5 out. The other 6 must come from squares 1 to 3, and 1 + 2 + 3 is the only way, so all three are shaded.
Step 3: rows 1, 2 and 4 Row 1 has 4 + 5 = 9 and needs 1 more, so square 1. Row 2 has 5 and needs 1 more, so square 1. Row 4 has 4, with square 5 crossed out, and needs 6 from squares 1 to 3, which takes all three. 1 2 3 4 5 1 2 3 4 5 10 6 3 10 7 10 7 9 10 3
Rows 1, 2 and 4 each have one combination left.

Step 4: column 1, then rows 3 and 5

  • Column 1 needs 10 and has rows 1, 2 and 4, which make 7. It needs 3 more: row 3 is shaded and row 5 is out.
  • Row 3 now has square 1 and needs 2 more: square 2 is shaded and square 3 is out.
  • Row 5 has square 4 and needs 3 more. Square 1 and square 5 are out, so it’s square 3; square 2 is out.

Check the two columns we haven’t used: column 2 has rows 3 and 4 (3 + 4 = 7) and column 3 has rows 4 and 5 (4 + 5 = 9). Every target is met.

Step 4: column 1, then rows 3 and 5 Column 1 has rows 1, 2 and 4, making 7, and needs 3 more: row 3. Row 3 then has square 1 and needs 2 more: square 2. Row 5 has square 4 and needs 3 more: square 3. Every target is met. 1 2 3 4 5 1 2 3 4 5 10 6 3 10 7 10 7 9 10 3
The solved puzzle. Column 1 finished it off, then rows 3 and 5.

When you get stuck

  • Recount what’s left. Subtract the shaded squares from each target and look again at what the remainder allows. Most missed deductions are hiding in a line you looked at before it changed.
  • Check both directions. After each move, look at the crossing line, not just the next line along.
  • Go for the fewest combinations. A line with one or two combinations left is closer to giving something up than one with three.
  • You never need to guess. Every puzzle on PeerPuzzles can be solved with these techniques alone, so there is always a next step somewhere.

Next steps

Questions

Do I need to memorise the combinations?

No, but it helps. Working them out takes seconds in a 5×5 grid, and after a few puzzles the common ones stick on their own. Keep the table on this page open while you learn.

What if two combinations still fit a line?

Leave that line and look at the lines crossing it. Each square they settle rules out combinations here. In a puzzle with one solution, some line always has a deduction left; on PeerPuzzles that’s guaranteed, because every puzzle is checked to be solvable without guessing.

Is there a trick for big grids?

Count the empty squares instead of the shaded ones whenever a target is more than half the largest. A 9×9 row with target 40 is just a row whose empty squares add up to 5, which is much easier to think about.

Why does a middle target tell me nothing?

Because its combinations between them use every square, and leave every square out. In a 5×5 grid that’s true of every target from 5 to 10. Those lines only open up once their crossing lines have settled a square or two.