How to Solve a Bridges Puzzle

One real 9×9 with fifteen islands, from an empty board to a finished network. Every move below is the move the solver made, in the order it made it.

The order to look in

Start by finding the islands with no choice. Any island whose number equals twice the number of directions it can see is completely decided — a 4 with two neighbours, a 6 with three, an 8 anywhere — and so is any island with a single visible neighbour, whatever its number. These cost nothing to spot and they are what opens a board.

Then take the other half of the same idea: any island that already has all the bridges its number asks for is finished, and every remaining link touching it can be ruled out. This is the step people skip, and skipping it is why boards feel stuck. Ruling a link out is not visible progress — nothing new appears on the board — but it takes room away from the island at the far end, and that island very often becomes decided as a result.

Only when both of those run dry is it worth doing arithmetic. Weighing one link against the rest of an island’s links is real work compared with glancing at a number, and doing it early means weighing links that a free sweep was about to settle anyway. The rhythm that finishes boards is: sweep for forced islands, close out the finished ones, sweep again, and only then start counting.

The board, move by move

The board below is a 9×9 with fifteen islands, graded hard — chosen not because a beginner should start there but because its first five moves happen to use four different arguments, one after another, which no easier board does. The four positions shown are the four moments where something new is being used; the eighteen moves in between are more of the same.

Read each pair of boards left to right. On the left is the position as it stood: the islands the argument looked at are marked, and the link it decides is outlined. On the right is the same position a moment later. A faint dashed line is a link that has been ruled out — that is a real result and is often the whole content of a move, which is why it is drawn rather than left blank. A hairline is a link nobody has decided yet.

1. The 2 at row 5, column 2 has exactly one island in its line of sight — the 4 out to its right. Everything it owes has to go that way, so the link is doubled before anything else on the board has been looked at.

   Full house

2. That double bridge now runs across column 3, and the 3 at row 4 and the 4 at row 7 were looking at each other down it. They cannot be joined any more, so the link is struck out — nothing is built, but two islands just lost a direction.

   Blocked crossing

3. The 2 at row 7, column 5 was filled by the previous move and has both the bridge ends it asked for. Its remaining link upwards is therefore closed, which is the bookkeeping step people skip and then wonder why the board stops moving.

   Spent island

4. Now the free moves run out and the arithmetic starts. The 3 in the top-right corner can only reach two islands, and the one below it can supply at most two bridge ends — so at least one has to come along the top row.

   Spare capacity

what the reasoning read what it decides what it wrote

Eighteen more moves of the same four ideas, and every island is satisfied. Here is the finished network.

When the board will not move

Re-sweep before concluding anything, and start with the islands nearest your last bridge. By far the commonest cause of a stuck board at this level is an island that your own previous move finished, whose remaining links you have not closed out. Closing them is free and it usually unlocks two or three more islands in a chain.

If that fails, go round every island that still has room and ask the arithmetic question about each of its links in turn: if every other link from this island carried as much as it possibly could, would this one still have to carry something? That is a slow question and it is worth asking exhaustively rather than where you suspect, because an island that looks unpromising and one that is about to give way look identical until you do the sum.

And if a full sweep and a full round of arithmetic both come up empty, the board wants the connectivity argument: look for a link whose use would close a small group off from the rest of the board. What the board does not want, ever, is a guess — every one published here was checked to be finishable without one.

Common questions

More Bridges pages

• Hashi puzzleUnlimited boards
• Daily bridgesA new board every day
• Bridges rulesFive rules, in full
• Bridge techniquesThe whole ladder
• Island countingTechnique — easy
• CrossingsTechnique — medium
• Spare capacityTechnique — hard
• IsolationTechnique — expert
• Guessing at bridgesTechnique — refused, here too
• Printable bridgesFor paper
• Bridges archiveEvery past board