Bridges

A bridge is a cell where one path can pass straight across from left to right while another passes straight through from top to bottom. The two paths cross without touching. Everywhere else the normal rules hold. Paths don't cross, and every cell gets filled.

On these diagrams a bridge is drawn as a dashed square. The solver at /solve/ doesn't read bridges yet. Every board here was checked with the site's Python solver, which models a bridge as one cell with two separate lanes.

How a bridge works

A bridge has four exits, and they come in fixed pairs. A path that enters from the left leaves to the right. A path that enters from the top leaves from the bottom. Nothing can turn on a bridge, so a path can't come in from the left and leave through the top.

That gives you three ways to think about a bridge:

• It's two lanes in one cell. The horizontal lane and the vertical lane are separate, and each can hold a different color. In the site's solver a bridge counts as filled once at least one path uses it, and one color can't use both lanes, so a path never crosses itself there.
• It's a jump. A tip next to a bridge that moves into it comes out on the far side in the same move. When you count a tip's open neighbors, a bridge beside it only counts as open if that lane is free and the cell beyond the bridge is empty or is that color's partner.
• It's the only place two paths can cross. If one color's route has to separate another color's two dots, the two colors have to cross at a bridge, one in each lane.

A bridge on the edge of the board loses one lane, since the lane pointing off the board leads nowhere. It behaves like an ordinary cell that only allows a straight path along the edge.

A worked example

Red starts with a forced move. Red's dot at row 1, column 4 sits between Blue's dot and Orange's dot on the top edge, so its only exit is down to row 2, column 4.

Green has to use the bridge. Look at Green's dot at row 2, column 3. Above it is Blue's dot, to its left is Red's dot, and to its right is the Red path that just arrived. Its only way out is down into the bridge. A path that enters from the top leaves from the bottom, so Green comes out at row 4, column 3, right beside its partner. Green now holds the bridge's vertical lane.

Red has to cross Green. Red needs to get from row 2, column 4 to its partner at row 2, column 2, and Green now runs down column 3 from row 2 to row 4 in between. Red has two options. It can cross the bridge along row 3, using the free horizontal lane, or it can go all the way around below Green through row 5.

Going around fails. That route would run from Red's dot on the top edge down to the bottom edge without touching the bridge, and it would split the board in two. Orange's dot at row 1, column 5 would be on one side and its partner at row 5, column 1 on the other, with no bridge between them to cross on. This is a bottleneck found by thinking ahead. So Red runs down to row 3, column 4, straight across the bridge to row 3, column 2, and up to its dot.

The rest is forced. Blue's dot at row 1, column 3 can now only go left. It follows the top-left corner and the left edge down to its partner at row 4, column 2. Green joins its partner directly. Orange takes the whole right edge and the bottom row.

Replace the bridge with a plain cell and the board has no solution. Green still has to go down into the center cell, Red still can't get around Green without cutting Orange off, and one ordinary cell can't hold both of them.

Tactics

Pair the exits. Before you start, note which cells sit on each side of every bridge. Left pairs with right and top pairs with bottom. If a color is on the left side and the right-side cell is taken by a different color's finished path, that color can't use the horizontal lane.

Find the crossing pairs first. Look for two colors whose dots alternate around a region, so that any route for one has to cut between the other's dots. Those two colors must cross, and on a bridge board they can only do it at a bridge. Then decide which one goes across and which goes down. Usually one of them is pushed into a lane by a forced move, as Green was here, and the other gets the remaining lane.

A wall with a bridge in it has a hole. If a finished path runs through a bridge, the other lane is still open, so the path isn't a solid wall there. A color can cross it at the bridge, moving at right angles to it. Before you decide a color is cut off, check every bridge along the wall.

Check the cell beyond. Forced moves next to a bridge are easy to misread. A tip whose only open neighbor is a bridge looks forced, but if the cell on the far side is taken, that lane leads nowhere. The tip is really boxed in, and an earlier move was wrong.

Parity is off. The checkerboard count assumes each cell holds one path. A bridge can hold two, so don't use the count on bridge boards.