Draw walls along the grid lines to divide the whole board into regions called galaxies. Every cell belongs to exactly one galaxy.
Every galaxy contains exactly one dot, and every dot belongs to exactly one galaxy.
Every galaxy is unchanged by a 180° turn around its own dot: if a cell is in a galaxy, so is the cell directly opposite it through that dot.
Every galaxy is a single connected region — you can walk between any two of its cells through shared sides.
A dot always belongs to the galaxy it sits in: a dot on a border joins the two cells it touches, and a dot on a corner joins all four.
The puzzle is solved when every cell lies in a galaxy that holds exactly one dot and is symmetric around it. Every published Galaxies puzzle has exactly one valid solution.
Two dots can never share a galaxy, so the line between two cells that different dots sit on is always a wall. Sweep the board for these before anything else — they cost no thought and they seed everything that follows.
Example: Two dots sit at the centres of side-by-side cells. Draw a wall on the line between those two cells straight away.
A galaxy looks the same after a 180° turn about its dot, so a wall on one side of the dot has a twin directly opposite it. Whenever you commit a wall around a dot, turn it through that dot and draw the partner too.
Example: A dot sits at a cell centre and you wall off the cell to its left. Turn that wall about the dot and wall off the cell to its right as well.
A cell can only join a dot if the cell directly opposite it through that dot is still on the board. Near an edge or a corner most dots fail that test, which often leaves just one dot that can claim the cell.
Example: The top-left cell can only belong to a dot in the top-left area of the board — any dot further away would put its partner off the grid entirely.
Work out which dots could reach a given cell at all, remembering that the path must stay inside that dot’s own galaxy. When only one survives, the cell is settled, and so is the cell opposite it through that dot.
Example: A cell is boxed in by walls on three sides and only one dot lies beyond the fourth. That cell joins that dot, and its partner about the dot joins too.
A dot at a cell centre whose four neighbours are all claimed by other galaxies is a galaxy of one cell. That is perfectly legal and very common — wall it in on all four sides and move on.
Example: A lone dot sits in a cell surrounded by cells already settled into other galaxies. Wall all four of its sides.
A cell and its partner about a dot join that galaxy together or not at all. So if you can prove one of the pair is impossible, the other is out too — which is often easier to see from the far side.
Example: The cell opposite a candidate already belongs to another galaxy, so the candidate cannot join this dot either.
A region with no dot in it turns red — it can never reach one, because walls fully enclose it. A region with one dot goes green when it is symmetric and red when it is not. A region with two or more dots is simply unfinished and stays plain.
Example: You wall off a pocket and it turns red. Nothing you add later can rescue it, so remove one of the walls around it now.
The board is done when every region is green. A region that is neither green nor red still holds two or more dots, so it is the one that needs splitting — no counting required.
Example: Every region is green but one plain pocket remains — it holds two dots, so look for the wall that separates them.