How to solve Hashiwokakero (bridges) puzzles
A bridges puzzle (Hashiwokakero, or Hashi) looks like a maze but is pure counting. Every island's number is a budget, every bridge spends from two budgets at once, and a few rules tell you which bridges are certain. Here are the techniques that solve every Causeway puzzle, in the order you will use them.
The rules in one paragraph
Islands are circles holding a number. A bridge joins two islands in a straight horizontal or vertical line, with no island in between. At most two bridges join the same pair, bridges never cross, and when you finish, the number of bridges at each island equals its number and every island can be reached from every other. You can try each technique below on the daily Causeway puzzle, or paste a board into the bridges solver to see the same reasoning applied step by step.
1. Count the exits
For each island, list the neighbours it can see in a straight line, and think of each as a place that takes 0, 1 or 2 bridges. If the number equals twice the neighbours, every link is a double. A 4 in a corner has two neighbours, so both links are doubles. A 6 on an edge with three neighbours, and an 8 anywhere, are always full doubles. The same idea gives half-certainties: if the number is one less than that maximum, every link carries at least one bridge. A 7 with four neighbours must send at least one bridge to each of them. More generally, list the ways the number can be split among the links; a bridge that appears in every split is certain, and a link that gets nothing in any split is empty.
2. Take away what is already spent
Once an island has all the bridges it needs, it takes no more, so every other link out of it is empty. Cross those links out in your head and recount the neighbours of the islands on the far side: they often now have fewer exits, and technique 1 starts working again. Work like this, around and around, and most of a puzzle falls out.
3. Crossings
If a bridge is certain, it blocks every line that would cross it. Those blocked links count as dead ends for the islands at their ends. Look for an island whose only way to reach its number was across a certain bridge: it has just lost an exit.
4. The loop rule
This is the one that separates easy puzzles from hard ones. The finished network must be connected, so you may never complete a group of islands whose numbers are all met while other islands remain outside it. If giving a link a bridge would finish such a group, that link cannot have the bridge, or cannot have as many. The smallest cases are the famous ones: a 1 can only have one bridge, so two 1s that can see each other must not be joined, because they would be a closed pair cut off from everything else. In the same way, two 2s joined by a double bridge would be a finished, isolated pair, so that link carries at most one bridge. Look at other small clusters too, and ask whether they could be sealed off. Causeway's Thursday, Saturday and Sunday puzzles are the ones that need this rule; Monday, Tuesday, Wednesday and Friday can be done with techniques 1 to 3. Both statements are checked for each puzzle by the program that makes them: it solves each with and without the loop rule.
When you are stuck
Re-read each island and count the exits again, because earlier moves change the counts. Look for the island with the fewest spare bridges left rather than the biggest number. Use Check to find a wrong bridge (it adds a check to your result), or the solver to see which rule comes next. A Causeway puzzle is built to be solved without trying a bridge to see what happens, so if you are guessing, a deduction is hiding somewhere.