Aquarium
Play Aquarium — water finds its own level in each tank. One deduction settles six more cells, which is why row and column totals are enough.
Click a cell to raise that tank's water to its row; click a filled cell to drain to just below it. Water is always level and always sinks, so every position you can reach is a legal one — the puzzle is working out where each waterline sits. The numbers give filled cells per row and column. Everything runs in your browser; nothing is uploaded.
One cell tells you 6.0 more
Work out that a single cell holds water and you have settled a good deal more than that cell. Water is level, so every cell of the same tank on that row is filled too; water sinks, so every cell of that tank below is filled as well. Across 40 random layouts per size, that is what one deduction is worth:
| Board | Cells also settled | Most | Settles nothing else | Settles five or more |
|---|---|---|---|---|
| 6 × 6, 6 tanks | 3.54 | 12 | 6.1% | 33.2% |
| 8 × 8, 8 tanks | 4.94 | 15 | 4.1% | 51.8% |
| 10 × 10, 10 tanks | 6.04 | 17 | 3.0% | 62.1% |
On a 10×10 only 3.0% of cells settle nothing but themselves, and 62% settle five or more. That is the feeling of an Aquarium coming apart in your hands once the first tank gives way.
A tank is one number, which is why the totals are enough
Here is the thing that makes Aquarium different from most grid puzzles. A tank does not have a state per cell. Its entire state is one number — where the waterline sits. A tank spanning five rows has six possible states, not two-to-the-power-of-its-cells.
So the number of legal fillings is a product of small numbers, and it is tiny next to the number of ways to shade cells if there were no water rule at all:
| Board | Ways to shade cells | Legal fillings | Removed by the water rule |
|---|---|---|---|
| 6 × 6 | 6.87e+10 | 2.95e+3 | 2.3e+7× |
| 8 × 8 | 1.84e+19 | 1.16e+5 | 1.6e+14× |
| 10 × 10 | 1.27e+30 | 7.00e+6 | 1.8e+23× |
On a 10×10 the water rule cuts 1.27e+30 shadings down to about 7.0 million legal fillings — a factor of 1.8e+23 — before a single clue has been read. Twenty numbers round the edge are then plenty to finish the job, which is why a random filling is pinned by its own totals essentially every time.
And it really is the water rule, not the numbers
"Row and column totals pin the answer" would be a far less interesting claim if the totals were doing it on their own. They are not — and the way to show that is to take the water rule away by degrees. Keep the grid, keep the totals, and shrink the tanks. Smaller tanks are shorter, and a shorter tank has less to say:
| Tanks on a 6×6 | Mean tank height | Puzzles pinned | Mean solutions | Worst |
|---|---|---|---|---|
| 2 | 5.21 | 100% (150/150) | 1.00 | 1 |
| 3 | 4.33 | 100% (150/150) | 1.00 | 1 |
| 4 | 3.78 | 99% (149/150) | 1.01 | 2 |
| 6 | 3.06 | 100% (150/150) | 1.00 | 1 |
| 9 | 2.36 | 97% (146/150) | 1.03 | 2 |
| 12 | 1.95 | 89% (134/150) | 1.12 | 3 |
| 18 | 1.50 | 43% (65/150) | 2.75 | 28 |
| 24 | 1.25 | 10% (15/150) | 26.48 | 314 |
| 30 | 1.10 | 0% (0/150) | 232.21 | 500+ |
| 36 | 1.00 | 0% (0/150) | 478.89 | 500+ |
The collapse is sharp and it lands exactly where tanks stop being tall. At a mean height of about two rows, 89% of puzzles are still pinned. Drop to 1.50 and it is under half. At one cell per tank — where the water rule has nothing left to say, every cell being an independent yes or no — not one puzzle in 150 is pinned.
That bottom row is not a quirk of this page. With one cell per tank the puzzle is exactly the problem of rebuilding a grid from its row and column sums, which has been known to admit many answers since the 1950s: any two filled cells at opposite corners of a rectangle can swap for the other two corners without changing a single total. The totals were never the reason.
How to use
- Pick a board size and press New puzzle.
- Click a cell to raise that tank’s water up to its row.
- Click a filled cell to drain the water back to just below it.
- Match the number beside each row and above each column — they count filled cells.
- Water is always level and always sinks, so every position you can reach obeys the rules.
Frequently asked questions
What are the rules of Aquarium?
The grid is divided into tanks. Water finds its own level in each tank, so the filled cells of a tank are exactly those at or below one waterline — a filled row of a tank is filled right across it, and everything below is filled too. The numbers beside each row and above each column say how many cells in that line hold water.
How much does one deduction tell you?
On a ten by ten board with ten tanks, working out that one cell holds water settles about six more on average, and 62% of cells settle five or more. Only 3% settle nothing beyond themselves. Water being level fills the rest of that row within the tank, and water sinking fills everything below.
Why are row and column totals enough to solve it?
Because a tank has hardly any freedom. Its entire state is one number — where the waterline sits — so a tank spanning five rows has six states rather than two to the power of its cells. On a ten by ten that cuts about 1.27e30 ways of shading cells down to roughly seven million legal fillings, a factor of 1.8e23, before a single clue is read.
Is it the water rule or the numbers doing the work?
The water rule, and you can watch the difference appear. Keep the grid and the totals and shrink the tanks: with tanks averaging five rows tall every puzzle is pinned, at two rows 89% are, and at one cell per tank not one puzzle in 150 is. A one-cell tank has no water rule left to apply.
What happens at one cell per tank?
The puzzle becomes the problem of rebuilding a grid from its row and column sums, which has been known since the 1950s to admit many answers. Any two filled cells at opposite corners of a rectangle can swap for the other two corners without changing any total, so the totals alone can never pin it.
Are Aquarium puzzles always solvable without guessing?
The ones generated here always have exactly one answer, which is checked before the puzzle is handed to you. At the standard sizes a random arrangement of waterlines is pinned by its own totals essentially every time, so unlike most grid puzzles there is barely any searching involved in making one.
Does this send anything anywhere?
No. Puzzles are generated and solved entirely in your browser, and nothing is uploaded.
🔒 This tool runs entirely in your browser. Nothing you enter is uploaded, logged, or stored.