Hashi Puzzle

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Five rules, and only the last one is unusual

Numbered islands sit on a rectangle. Join them with bridges so that every bridge runs straight, horizontally or vertically, between two islands and passes over no island on the way; no pair of islands carries more than two bridges; no bridge crosses another; and every island ends up with exactly as many bridge ends touching it as its number says. Four rules, all of them local, all of them obvious the moment you have drawn one bridge.

The fifth is the one that makes the puzzle. When you are finished, the whole thing has to be a single connected network — you must be able to walk from any island to any other along the bridges you have drawn. Nothing about a particular island tells you whether you are obeying it, and it is entirely possible to satisfy every number on the board and still be wrong because you have built two separate islands-and-bridges continents.

That rule is also where the interesting reasoning lives. Most of what an experienced solver does late in a board is not arithmetic at all; it is noticing that a certain bridge would seal a handful of islands off from everything else, and that therefore it cannot be drawn.

What the numbers are, and what they are not

An island’s number counts bridge ends, not neighbours. A 4 might be joined to four islands by one bridge each, or to two islands by two bridges each, or to three islands with one of them doubled. Reading the numbers as a count of connections is the single commonest beginner error and it makes half the board look impossible.

Eight is the largest number that can appear, because an island has at most four directions and each direction carries at most two bridges. A 8 is therefore completely decided the instant you see it, and so is any island whose number equals twice the number of directions it can actually see — a 6 on an edge, a 4 in a corner. Finding those is the whole of the opening move.

The numbers are not clues in the sense that a binary puzzle’s symbols are clues. Nothing is hidden here: the entire puzzle statement is visible from the start, and what is unknown is not any island’s value but how much traffic runs along each possible connection. That difference changes how the solver works and it changes how you should look at the board.

Where the difficulty comes from

Not from the number of islands. A 15×15 board with forty islands and no bottleneck is a long puzzle rather than a hard one — it asks the same two questions forty times. The four levels here name the reasoning a board forces instead.

Easy is finished by looking at one island at a time: the ones with no choice, and the ones that are already finished and can therefore close everything else they touch. Medium adds the crossing rule as a working tool rather than a constraint you obey. Hard makes you weigh an island link by link, asking what the others could carry so as to learn what this one must. Expert is where connectivity decides it — where a bridge is ruled out only because drawing it would strand a group.

The grade is measured, never estimated. Every published board is solved by a program that applies the cheapest technique that fits, over and over, and the hardest one it was ever forced onto is the level printed on the board. A board built for one level that turns out to need another ships under the level it measured, which occasionally means asking for expert on a small board and being handed a hard one — an honest label beats a flattering one.

Common questions

More Bridges pages

• Daily bridgesA new board every day
• Bridges rulesFive rules, in full
• Solving a boardA 9×9, worked
• 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