Kakurasu Puzzle

Kakurasu is a shading puzzle. You mark every cell shaded or empty so the shaded cells in each row add up to the number on the left, and the shaded cells in each column add up to the number on top. You can play grids from 5×5 up to 9×9, and every puzzle here has exactly one solution that you can reach by logic, with no guessing.

Rules

The board is an n×n grid. A number sits to the left of every row and above every column. Those are the sums you have to hit.

Along the bottom, under the columns, the weights run 1, 2, 3, and so on, starting at the left. Down the right side, beside the rows, the weights run 1, 2, 3, and so on, starting at the top. Those numbers are the weights. They are not extra sums, and they do not change.

Shade a cell or mark it empty. A shaded cell in a row adds the weight of its column. A shaded cell in a column adds the weight of its row. An empty cell adds nothing.

The puzzle is solved when every cell is shaded or empty, every row matches the sum on its left, and every column matches the sum above it.

Controls

Choose Shade or Cross under the board. Tap a cell, or drag across cells, to place that mark. Tap a cell that already has the mark to clear it. A drag follows the first cell: it places the mark, or it clears when that cell already has it. With a mouse, right-click places a cross, or clears a cross that is already there.

On a keyboard, the arrow keys move around the grid. S or 1 shades the cell. E, X, or 2 marks it empty. Backspace, Delete, or 0 clears it. Space or Enter places the selected mark, or clears it when the cell already has that mark.

A sum turns teal when the shaded cells already match it, even if you still have cells to mark empty. A sum of 0 stays plain on an empty line until you mark that line or every cell on it is empty. A sum turns red when the shaded weight has gone past it, or when the cells still open cannot reach it. Press Hint for one step the rules already force, with a sentence saying why. Press Check when you want the board checked against the rules. Undo and Redo step through your moves, and Reset clears the board. When every cell is shaded or empty and those marks match the unique solution, the board turns teal.

Techniques

Easy grids follow from one line at a time:

• Zero line: a sum of 0 means every open cell on that line is empty.
• Full line: a sum that equals every weight on the line means every open cell is shaded.
• Single weight: only one open weight equals what the line still needs, so that cell is shaded and the other open cells are empty.
• Too big: an open weight larger than the sum still needed is empty.
• Cannot skip: the line cannot reach its sum without these cells, so they are shaded.
• Already enough: the shaded cells already match the sum, so the open cells are empty.
• Rest of the line: the open weights add up to exactly what is left, so those cells are shaded.

Medium grids also look at every way one line can add up, and keep the cells that every way agrees on.

Hard grids look across lines. They try the completions of one line that still fit the other sums, and they try one cell both ways when a single careful check is enough.

Beginner tips

Start with the 0 sums and the sums that use every weight. A 0 clears the whole line. A sum that equals 1 + 2 + … + n shades the whole line.

When a line is already at its sum, mark the rest of that line empty before you look for the next shade. Those empty marks often force the crossing line.

If you are new to Kakurasu, start with the daily 6×6 easy grid, or a 5×5 easy grid, and move up once zero lines and full lines feel natural.