Bridges: try both ways

When counting and the network rule both stall, pick one bridge, suppose it goes one way, and follow the rules. If they break, it goes the other way.

Practice now

Three bridges in, and counting and the network rule both stall. The 1 in the top row could join the 3 on its right or the 3 below it.

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How it works

Every hard Bridges map reaches a point where no island can be counted out and no group is about to be cut off. Then solvers try a bridge both ways.

Pick a pair of islands that could share a bridge. Suppose they do, and follow the easy rules from there. If an island ends up short, or a group is cut off, the supposition was false. That is a proof by contradiction, and it settles the pair.

Which bridge to try

Try a pair where either answer changes a lot: a 1 with two neighbors, or a bridge that would fill a small group. The shorter the chain of rules that follows, the easier it is to check.

Worth memorizing

Suppose one way and follow the rules.
A broken rule proves the other way.
Try small numbers first.
They fill up fast, so the chain is short.

Practice

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Question 1

Suppose

Suppose the 1 joins the 3 on its right. What goes wrong once you follow the counting?

Question 2

So which island does the 1 in the top row join?

Where you’ll use this

Proof by contradictionThe logic behind it No island cut offThe rule that usually breaks Play BridgesWeekend maps need this

Play the hard puzzle of the weekPut this lesson to work Next: Fillomino: growing regionsA region that still needs cells must grow, and when it has one open side, it grows there. A finished region bars its number from every neighbour.