Kakurasu Puzzle & Clue Maths
Shade cells so every row and column hits its number — and see why exactly six clues out of the whole range settle a line on their own, at any size.
Shade cells so each row and column hits its number. A shaded cell counts as its position — first cell 1, second 2, and so on. Click to shade.
Exactly six clues pin a line down — always six, at every size
A clue is a subset sum of 1…n, so the question is how much one clue tells you on its own. Almost nothing — and it does not improve with size. The clues with exactly one possible line are 0, 1, 2 and the top three, and that is the whole list at every width. A clue of 3 is already ambiguous: it is both {3} and {1,2}.
| Width | Possible clues | Decisive on their own | Share | Worst clue admits |
|---|---|---|---|---|
| 4 | 11 | 6 — 0, 1, 2, 8, 9, 10 | 55% | 2 lines |
| 5 | 16 | 6 — 0, 1, 2, 13, 14, 15 | 38% | 3 lines |
| 6 | 22 | 6 — 0, 1, 2, 19, 20, 21 | 27% | 5 lines |
| 7 | 29 | 6 — 0, 1, 2, 26, 27, 28 | 21% | 8 lines |
| 8 | 37 | 6 — 0, 1, 2, 34, 35, 36 | 16% | 14 lines |
So a bigger Kakurasu is not a bigger version of the same puzzle. The share of clues that say anything by themselves collapses from 55% to 16%, and everything has to come from rows and columns disagreeing with each other.
And the middle clues are hopeless
How many different lines make each clue on an 8-wide board:
The clue 18 admits 14 different lines. Nothing about it narrows anything until a column disagrees. Only the six green bars at the ends settle a line by themselves.
The counts are exactly symmetric, and for a reason worth seeing: swap every shaded cell for an unshaded one and a line summing to k becomes a line summing to the total minus k. So the number of lines making 5 equals the number making 31 on an 8-wide board — necessarily, not by coincidence.
How to use
- Shade cells so each row adds up to the number beside it.
- A shaded cell counts as its position — first cell one, second two.
- Do the same for every column.
- Green clues are the ones that settle their line by themselves.
Frequently asked questions
What are the rules of Kakurasu?
Shade any cells you like, so long as each row adds up to the number beside it and each column adds up to the number below it. A shaded cell counts as its position in that line, so the first cell is worth one, the second two, and so on.
Which Kakurasu clues tell you the answer on their own?
Exactly six of them, at every board size: zero, one, two, and the top three. Zero means an empty line, one means the first cell, two means the second. A clue of three is already ambiguous because it is both a single three and a one plus a two.
Do bigger Kakurasu boards give you more to go on?
The opposite. The number of decisive clues stays at six however wide the board gets, while the range of possible clues keeps growing, so the useful share collapses from fifty-five per cent on a four-wide board to sixteen per cent on an eight-wide one.
How ambiguous can a single clue be?
On an eight-wide board the clue eighteen admits fourteen different lines. Nothing about that clue narrows anything at all until a column disagrees with it, which is why the puzzle has to be solved from rows and columns pulling against each other.
Why are the clue counts symmetric?
Because swapping every shaded cell for an unshaded one turns a line summing to k into a line summing to the total minus k. So on an eight-wide board the number of lines making five is necessarily the same as the number making thirty-one.
Is Kakurasu the same as Kakuro?
No. Kakuro is a crossword of digits that add to given totals. In Kakurasu you shade cells rather than write numbers, and a cell contributes its position rather than a value you choose, which makes every clue a subset sum of one to n.
Does every puzzle here have one answer?
Yes. Each generated puzzle is checked by counting its solutions and thrown away unless there is exactly one, rather than assuming the grid it came from was the only possibility.
Does this send anything anywhere?
No. The puzzles are generated and checked entirely in your browser.
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