Bridges
Join the islands so each one has exactly its number of bridges.
The maths Vertex degree in a graph
8 islands still need bridges. The network is in 8 separate pieces.
How to play
Every island carries a number. That number is how many bridge-ends must meet there: an island marked 3 needs three bridges touching it, counting a double bridge as two.
Bridges run straight across or straight down, never diagonally. A pair of islands may be joined by one bridge or two, and no more. A bridge may not cross another bridge or pass through an island, and when you are finished every island must be reachable from every other one along the bridges.
There may be more than one way to satisfy the numbers. Any arrangement that meets all four rules counts as solved, not only the one the puzzle was built from.
An island shows a second ring around its number once it has exactly the bridges it needs, and a dashed ring if it has too many. A faint dotted line marks every place a bridge is allowed to go, and a cross marks a place another bridge already runs through.
Mouse or touch: click the dotted line between two islands to add a bridge, click again for a double, again to take it away. You can also click one island and then a second island in line with it.
Keyboard: Tab reaches the board, then the arrow keys move between islands. Press Enter or Space to pick the island you want to build from โ its ring turns dashed. With an island picked, an arrow key builds toward the neighbour that way, a second press of the same arrow makes it a double, a third takes it away. Press Escape, or Enter on the same island, to let it go and move around again. Home and End jump to the first and last island.
Why this is mathematics
The number on an island is its degree โ how many edges must meet there. Building the network means satisfying every degree at once while keeping the whole thing connected and never crossing. It is the same bookkeeping that underlies road networks and circuit layout.