Capitol Magnets

• Split the grid into states: connected groups of squares, each symmetric under some rotation or reflection.
• A state has a capitol when every non-trivial symmetry of it fixes exactly one square; that square is the capitol. A 1×3 bar and the X pentomino have none. A single square is its own capitol.
• Moving to an orthogonal neighbour costs the size of the state you are in. Crossing a border between two states costs the smaller of the two sizes.
• A number is the cheapest total cost from that square to any capitol, so a 0 marks a capitol.
• Answer: label every square with its state's size, sum each row, and add up the squares of the row sums.

Keys: 1–9 and 0 pick colors, C capitol, E eraser, Ctrl+Z undo. The 11×11 board is the original puzzle; Check tells you whether your grid satisfies every rule and clue.