Aquarium: read the bottom row first

Water settles, so the bottom row decides

Each outlined region is one aquarium with a single horizontal water level. Fill it and the water lies flat across the whole region at that height, then fills every row beneath. Nothing floats, and nothing half fills a row.

That gives you two deductions, and they point in opposite directions. Water in a cell makes every cell of the same tank at that height or lower water as well. Air in a cell makes every cell of the same tank at that height or higher air as well.

The second one is the workhorse. A tank's lowest row is wet whenever the tank holds anything at all, so proving that lowest row dry empties the entire tank in one move, however tall it is.

The board

Six aquariums on a six by six grid. Reading the row clues from the top: 0, 0, 2, 2, 4, 4. Reading the column clues from the left: 0, 1, 2, 2, 4, 3.

Two of those clues are zero, and the difference between them is the whole lesson. Zero on a row is worth six dry cells. Zero on column 1 is worth ten, because of which tank sits underneath it.

Start with the zeros, but weigh them differently

Rows 1 and 2 are clued zero, so all twelve cells in the top two rows are air. That is the ordinary reading of a zero and it costs no thought.

Column 1 is clued zero too, and it does far more. The tall aquarium on the left occupies column 1 in all six rows, which means its lowest row, row 6, includes the cell at row 6, column 1. If that tank held any water at all, its lowest row would be wet and column 1 would have a water cell in it. It cannot, so the tank is completely dry.

That is ten cells from one clue: column 1 in every row, and column 2 in rows 1, 3, 4 and 6. None of those cells were touched by the row zeros.

A saturated column gives the first water

Column 5 is clued 4. Its top two cells are already air from the row zeros, so four cells remain and all four must be water: rows 3, 4, 5 and 6 of column 5.

Rows 3 and 4 of column 5 belong to the large aquarium on the right. Water there means water across that tank at the same height, so row 3, column 6 and row 4, column 6 are water as well.

Row 4 is clued 2. Columns 1 and 2 are dry from the left tank, and columns 5 and 6 are now water, which meets the clue. So row 4, columns 3 and 4 are air.

One dry bottom empties a tank, one wet top fills one

Those two air cells are the lowest row of the small two by two aquarium in the middle. A dry lowest row empties the whole tank, so row 3, columns 3 and 4 are air too. Row 3 now reads two water cells, which matches its clue.

Column 2 is clued 1. Five of its six cells are already settled: four belong to the dry left tank, and the cell in row 2 is air from the row zero. The single water cell has to be row 5, column 2.

That cell is the top row of the lower middle aquarium. Water in a tank's top row fills the tank, so row 5, columns 2 to 5 and row 6, columns 3 to 5 are all water.

Row 5 is clued 4 and now has exactly four, so the single cell at row 5, column 6 is air. Row 6 is clued 4 and has three, so the single cell at row 6, column 6 is water. Every column total then checks out, and the board is done.

The mistake that costs the most time

Marking air at the top of a tall tank and then treating the rest of it as solved. Air high up only rules out the rows at that height and above, because water could still be sitting below it. Reading it the other way, and quietly emptying the tank, produces a board that contradicts itself several moves later, with nothing to show which move was wrong.

Say the direction out loud while you mark: air travels up, water travels down. The cell you want to find is a tank's lowest row, because that one cell decides the whole region.

The related trap is treating a bent aquarium as two tanks. A region that wraps around a corner still has one level, so a cell in the far arm at the same height as water you have already marked is water too, even with several dry cells of another tank in between.

Where Aquarium sits on HubbardGames

Aquarium is a timed game in the Library and in the Just for fun dailies. It is not one of the nine competitive games, so a finish is ranked on time against other players but pays no league points.

That makes it a good board to practise a technique on. Library sizes run from 6 by 6 up, and the small boards are where the bottom-row habit is quickest to build.

Tap a cell to cycle undecided, water and air, or select the Water or Air brush to paint one mark. When a row or column total turns green, tap the number to fill the rest of that line with air.