How to Play Kakurasu

Kakurasu (sometimes called “Index Sums”) is a binary-shading puzzle where every row and column has a target number, and where each cell silently carries a weight. Shade the right cells and the weights add up.

The grid is small, the rules fit on one line — but the deductions are deeply satisfying. Let's walk through one.

The Puzzle

Here is a 5×5 puzzle. The dark cells on the right and bottom are clues. Your job is to shade the right combination of white cells so that every row and column hits its clue.

The Weight System

Here's the trick that makes Kakurasu work. Every cell has a hidden weight:

• Inside a row, the columns weigh 1, 2, 3, 4, 5 from left to right.
• Inside a column, the rows weigh 1, 2, 3, 4, 5 from top to bottom.

The weight numbers are visualized below — the puzzle itself never shows them.

The Two Rules

Rule 1: Row clue = sum of column-weights of the shaded cells in that row

For each row, look at which columns are shaded and add up those column weights. The total must equal the right-side clue.

Rule 2: Column clue = sum of row-weights of the shaded cells in that column

Same idea, just rotated 90°. For each column, look at which rows are shaded and add up those row weights. The total must equal the bottom clue.

Magic Combinations

For a given clue, only certain subsets of weights can sum to it. When a clue has very few possible combinations — or when all combinations share a common weight — you can place cells with certainty.

For a 5×5 grid (weights 1, 2, 3, 4, 5):

Even when a clue has multiple combinations, the intersection of all those combinations gives you cells that are definitely shaded. Clue 11 above is a perfect example: every valid subset contains 2 and 5.

Walkthrough

Step 1 — Column 4 has clue 11

Look at column 4 with bottom clue 11. We need a subset of row weights {1, 2, 3, 4, 5} summing to 11. There are only two: {1, 2, 3, 5} and {2, 4, 5}. Both contain 2 and 5 — meaning column 4 row 2 and column 4 row 5 are definitely shaded, no matter which combination is correct.

Step 2 — Row 4 has clue 11

Same trick, applied to a row. Row 4's clue is 11. The subsets of column weights summing to 11 are {2, 4, 5} and {1, 2, 3, 5}. Both contain cols 2 and 5, so those cells are definitely shaded.

Step 3 — Column 2 has clue 6, and now we know one of its cells

Column 2's clue is 6. Subsets of {1, 2, 3, 4, 5}summing to 6: {1, 2, 3}, {1, 5}, {2, 4}. From step 2 we know row 4 in column 2 is shaded, so the chosen subset must contain 4. Only {2, 4} qualifies. Therefore column 2 = rows 2 and 4 — and every other row in column 2 is empty.

Step 4 — The rest cascades

Now constraints chain together. Tracking the deductions:

• Row 2 = {1, 2, 4}. Row 2's clue is 7. Subsets: {1,2,4}, {2,5}, {3,4}. We just placed col 2 in row 2 (from step 3), and we placed col 4 in row 2 (from step 1). Only {1, 2, 4} contains both. So row 2 also has col 1 shaded — and cols 3 and 5 empty.
• Column 5 = {1, 4, 5}. Col 5's clue is 10. Subsets: {1,4,5}, {2,3,5}, {1,2,3,4}. Row 4 in col 5 is shaded (from step 2) → eliminates {2,3,5}. Row 2 in col 5 is empty (just deduced) → eliminates {1,2,3,4}. So col 5 = {1, 4, 5}: rows 1, 4, 5 shaded.
• Row 5 = {4, 5}. Row 5's clue is 9. Subsets: {4,5}, {1,3,5}, {2,3,4}. Col 4 in row 5 shaded (step 1) eliminates {1,3,5}. Col 5 in row 5 shaded (just deduced) eliminates {2,3,4}. So row 5 = {4, 5}.
• Column 1 = {1, 2, 4}. Col 1's clue is 7. Subsets: {1,2,4}, {2,5}, {3,4}. Row 2 in col 1 shaded (from row 2 = {1,2,4}) and row 5 in col 1 empty (from row 5 = {4,5}) leave only {1, 2, 4}.
• Row 1 = {1, 4, 5}, Row 4 = {1, 2, 3, 5}, Row 3 = {4}. Each of these is now forced by the column placements above.

Tips for Beginners

• Look at extreme clues first. A clue of 0 means “no cells shaded”. A clue equal to the maximum (15 on a 5×5) means “every cell shaded”. Mid-extreme clues like 14 (only {2,3,4,5}) are also forced.
• Find the intersection of all valid combinations. Even when a clue has 2 or 3 valid subsets, weights present in every subset are forced. Clue 11 with subsets {2,4,5} and {1,2,3,5} forces 2 and 5.
• Cross-reference rows and columns. Once you place a single cell, it appears in both a row clue and a column clue — narrowing both at once. Most cascades start from a single placement.
• Track candidate sets, not individual cells. For each row and column, jot down its possible weight subsets. As cells get placed, eliminate subsets and the right answer appears.
• Don't guess. Every well-formed Kakurasu has a unique solution. If you can't see the next move, look for a row or column whose candidate list has shrunk to one option.

Ready to try one yourself? Hit the button below to play your first Kakurasu.