Tentai Show
Divide the grid into regions, one per dot, so that every region contains its dot, is connected, and looks exactly the same after a 180° turn about that dot. Dots sit on cell centres, edges, or corners. Click a dot to pick its galaxy, then click cells to paint them — the twin cell on the far side of the dot is painted for you, because it has no choice. Right-click erases.
One map is the whole puzzle
Cell (r, c), dot at doubled coordinates (cy, cx): the twin of the cell is (cy − r − 1, cx − c − 1). That single line covers dots on cell centres, on edges, and on lattice points — the doubled grid does not care which. Every rule in the solver is a sentence about a cell and its twin under that map, and that has a consequence you can watch in the hint button: the set of cells a galaxy might still own stays perfectly symmetric about its dot, through every rule in the ladder. Prune a cell on one side and the mirror rule takes the twin in the same round. Deductions here happen twice or not at all.
A rule you never have to write
A galaxy whose dot sits on an edge or a corner covers an even number of cells; a dot on a cell centre covers an odd number. That parity rule looks like a propagator worth having — and it can never fire. Once the candidate set is symmetric, undecided cells come in twins, and a twin contributes zero or two cells: every parity the rule could check is already correct. The parity rule is a theorem of the mirror rule, and the measured version of that sentence is in the stats: across 3 600 fixpoints, zero asymmetric candidate sets.
Connectivity is the rule that pays
Mirror alone finishes most 5×5 boards but falls off a cliff as boards grow: it never asks whether a cell can still walk home to its dot through surviving territory. The reach rule asks exactly that, and on 10×10 boards it is the difference between a third of boards solved and nearly all of them. The bridge rule sharpens it — when a pinned cell has a single corridor left to its dot, the corridor is taken — and the mirror invariant means bridges are found in pairs too, one on each side of the dot.