Counting moves: bridges, skyscrapers and lights

Bridges: capacity

An island can take at most two bridges to each neighbour it can see. If its number equals twice the number of neighbours, every connection is a double. If it is one less, every neighbour gets at least one bridge. A 1 with a single neighbour, a 2 with a single neighbour — these are free moves.

Bridges: no sealed groups

All islands must end up connected. So two islands that would together use up all their bridges on each other — a 1 next to a 1, a 2 next to a 2 joined twice — cannot be joined that way while other islands remain. The same reasoning applies to larger groups near the end of a puzzle.

Bridges: one step ahead

When counting stalls, pick an uncertain connection and imagine building it. If that immediately leaves some island unable to reach its number, or splits the map into two parts, the bridge is wrong and you can place the alternative. Hard and expert sheets need this at least once.

Skyscrapers: ones, Ns and the cap rule

A clue of 1 puts the tallest building next to it. A clue of N fixes the whole line in rising order. For the other clues use the cap: with a clue of c, the cell at distance d from that edge holds at most N − c + d. So a clue of 4 in a 6×6 grid allows at most 3 in the first cell, 4 in the second and 5 in the third.

Skyscrapers: height by height

Place the tallest buildings first, since they are visible from both sides. Then treat the next height as the tallest among what is left: it is hidden only by the buildings already placed in front of it. Opposite clues taken together often show exactly where it fits.

Lights: two facts

Pressing a lamp twice cancels out, and the order of presses does not matter. So a solution is simply a set of lamps, each pressed once. On the board sizes we use, that set is unique.

Lights: chasing

Go row by row from the top. For every lit lamp in the current row, press the lamp directly below it; that switches the lit lamp off without touching the rows above. When you reach the bottom, only the last row can still be lit.

Lights: reading the bottom row

What remains on the bottom row tells you which top-row lamps to press. The tables below were calculated for each board size: find every lit lamp of the bottom row (counted from the left), and press the top-row lamps listed for it. If a top-row lamp is listed an odd number of times in total, press it; if an even number of times, skip it. Then chase down again and the board goes dark.