How to Solve Aquarium: Water Levels, Clue Counting, and Tank Logic

Updated August 2026

Aquarium is a counting puzzle wearing a physics costume. The grid is divided into sealed tanks, and water in a tank always settles to one flat level — everything at or below the level is water, everything above is air. That single rule turns a grid of free cells into a much smaller set of choices, and the row and column clues do the rest.

Understand what "flat" actually forbids

Because a tank has one water level, two cells of the same tank in the same row are always the same: both water, or both air. That is the most useful consequence of the rule and the one beginners forget. The moment you determine one cell, you have determined every cell of that tank in that row — even cells far away horizontally, as long as the tank connects them.

Vertically, the rule is an implication: if a cell is water, every cell of the same tank below it is water too. If a cell is air, every cell of the same tank above it is air. Fill those consequences in immediately; they are free.

Work the extreme clues first

A row clue of 0 means the whole row is air, which by the vertical rule makes everything above it in those tanks air as well. A row clue equal to the row width means the whole row is water, which fills every tank cell below it. These full and empty rows and columns cascade a long way and cost no thought, so sweep them before anything else.

Count each row as an arithmetic problem

Within a row, each tank contributes either its full width in that row or nothing — there is no partial. So a row clue is a subset-sum problem over the tank widths present in that row. If a row has tanks of width 3, 2 and 1 and the clue is 4, the only combination is 3 + 1, so the width-2 tank is dry in that row and the other two are wet. Boards that look impossible usually have one row where the arithmetic has exactly one answer.

The same works negatively. If the clue is smaller than the narrowest tank in the row, that row is entirely air. If the clue equals the total width minus one and no tank has width 1, something is wrong — recheck your marks.

Use partial information to prune

Once some cells of a row are known, subtract them from the clue and re-solve the smaller problem. A row that was ambiguous with four unknown tanks often becomes forced once two of them are settled. Keep a light running count of "water still needed" and "cells still available" for every row and column, and revisit any line where those two numbers get close — when they are equal, everything remaining is water; when the needed count hits zero, everything remaining is air.

Let tanks talk across rows

A tank spanning several rows links their clues. If a tank must be wet in a low row, it is wet in every row below within the tank, which changes those rows' remaining budgets. Conversely, marking a cell dry high up in a tank forces air upward. Alternating between row arithmetic and vertical tank propagation is the rhythm of the whole puzzle.

A reliable order of attack

1. Fill every 0 row and column as air, and every full row and column as water.
2. Propagate: water flows down within a tank, air rises within a tank.
3. For each row, solve the subset sum over tank widths; mark anything forced.
4. Do the same for columns, remembering columns constrain single cells rather than tank widths.
5. Subtract known cells from each clue and re-solve the smaller problem.
6. Repeat from step two until the grid is full.

Every deduction here is arithmetic, so a stuck board is nearly always a row you have not re-counted since your last few marks.