Bridges Puzzle (Hashiwokakero)
A 3 can be more forced than a 6. The island number means nothing without its neighbour count, and 40% of combinations are forced.
A 3 can be more forced than a 6
Every guide to this game says to start with the 8s. That is true, and it is barely half the story, because the number on an island tells you nothing on its own. What matters is the number relative to how many neighbours it can reach. An island needing n bridges with k neighbours must send at least n − 2(k − 1) to each of them, since the other k − 1 can absorb only two apiece.
Work that through and the result is genuinely backwards: a 3 with two neighbours must send one bridge to each of them, while a 6 with four neighbours is not forced to send anything anywhere. The small number is solved on sight and the large one is not. Hunting for 8s and ignoring 3s is leaving the easiest deductions on the board.
Of the 20 possible combinations of island number and neighbour count, 8 — 40 per cent — force at least one bridge with no reasoning at all, and 4 force two. They are always the top two numbers for that neighbour count: 2k − 1 forces one bridge to every neighbour, and 2k forces two. Nothing below that is ever forced, which is where this differs from Kakuro — there the one-way clues sit at both ends of the range, and here only at the top. A low number can always be spread among its neighbours more than one way.
What is forced, by neighbour count
Bridges forced to every single neighbour, before any deduction.
| Neighbours | Island number | Forced to each | |
|---|---|---|---|
| 1 | 1 | 1 | one to every neighbour |
| 1 | 2 | 2 | fully forced |
| 2 | 1 | — | |
| 2 | 2 | — | |
| 2 | 3 | 1 | one to every neighbour |
| 2 | 4 | 2 | fully forced |
| 3 | 1 | — | |
| 3 | 2 | — | |
| 3 | 3 | — | |
| 3 | 4 | — | |
| 3 | 5 | 1 | one to every neighbour |
| 3 | 6 | 2 | fully forced |
| 4 | 1 | — | |
| 4 | 2 | — | |
| 4 | 3 | — | |
| 4 | 4 | — | |
| 4 | 5 | — | |
| 4 | 6 | — | |
| 4 | 7 | 1 | one to every neighbour |
| 4 | 8 | 2 | fully forced |
Read down the neighbour-count column rather than across: with two neighbours a 3 is forced, with three neighbours it takes a 5, and with four it takes a 7. The same number means completely different things in different places on the board, which is why counting a neighbour before reading the number is the habit worth building.
What if a pair could take three bridges?
The rule generalises exactly. What it costs you is not difficulty.
Replace the 2 with a cap m and the pigeonhole argument is unchanged: the other k−1 neighbours can absorb at most m apiece, so each one is owed at least n − m(k−1). The forced numbers are still exactly the top m for each neighbour count — and the share of forced pairs works out at mK ÷ mK(K+1)/2 = 2/(K+1), which has no m in it at all.
| Bridges allowed per pair | Pairs that force | Share |
|---|---|---|
| 2 | 8 of 20 | 40.0% |
| 3 | 12 of 30 | 40.0% |
| 4 | 16 of 40 | 40.0% |
| 5 | 20 of 50 | 40.0% |
So the 40% is a fact about a square grid having four directions, not about the two-bridge cap. Change the board instead of the cap and it moves; change the cap and it does not: 3 directions 50.0%, 4 directions 40.0%, 6 directions 28.6%, 8 directions 22.2% — the same at every cap in each case.
What a third bridge does cost is uniqueness. Generating layouts at random and counting solutions exhaustively — connectivity included — the share with exactly one answer falls by about a third, while the share that counting alone can finish barely moves:
| Board | One solution, cap 2 | cap 3 | Counting finishes it, cap 2 | cap 3 |
|---|---|---|---|---|
| 9×9, 9 islands | 71.5% | 57.0% | 92.7% | 94.3% |
| 9×9, 11 islands | 65.8% | 42.5% | 92.4% | 88.2% |
| 11×11, 13 islands | 57.3% | 35.0% | 92.1% | 83.3% |
The two columns move in opposite directions, which is the finding: a three-bridge variant is not harder to reason about, it is harder to pose. There is no separate three-bridge page on this site because the measurement said there was nothing extra to teach — the same pigeonhole, the same top-m structure, the same solving power. The last row is a smaller sample (120 layouts against 400), because well-posed layouts are rare enough there that the exhaustive count is slow.
How to use
- Click one island then another to lay a bridge between them.
- Click the same pair again for a double bridge, and again to remove it.
- Every island must end with exactly its number of bridges.
- Bridges cannot cross, and the finished map must be one connected whole.
Frequently asked questions
What are the rules of Hashiwokakero?
Join the islands with horizontal or vertical bridges so that each carries exactly the number written on it. At most two bridges may run between any pair, bridges may not cross one another, and when you are done every island must be reachable from every other. That last rule is the one people forget, and it is what stops a board full of satisfied numbers from being a solution.
Should I really start with the 8s?
Start with them, but do not stop there — the number alone tells you nothing. What matters is the number relative to how many neighbours the island can reach. An island needing n bridges with k neighbours must send at least n minus 2(k-1) to each one, because the other neighbours can absorb only two apiece. A 3 with two neighbours is forced; a 6 with four is not.
How can a 3 be more forced than a 6?
Because forcing is about the shortfall, not the size. A 3 with two neighbours has to put one bridge on each of them: two bridges to one neighbour leaves only one for the other, and zero to one neighbour cannot be made up by the remaining one. A 6 with four neighbours can be split many ways — two, two, one, one or two, two, two, zero and so on — so nothing at all is forced.
How much of a puzzle is forced from the start?
Of the twenty possible combinations of island number and neighbour count, eight force at least one bridge with no reasoning at all, and four force two. That is forty per cent of positions solved by counting rather than thinking, which is why scanning for them first is worth more than any other opening habit.
Which numbers are the forced ones?
Always the top two for that neighbour count. With one neighbour, a 1 and a 2; with two, a 3 and a 4; with three, a 5 and a 6; with four, a 7 and an 8. The higher of the pair is fully forced — two bridges to every neighbour — and the lower forces one to each. Nothing below that pair is ever forced.
Why is nothing forced at the low end?
Because a small number can always be spread among its neighbours more than one way. This is where Bridges differs from Kakuro, where the one-way clues sit at both ends of the range — the smallest sum for a run has to be 1+2+3 and nothing else. Bridges has no such floor, so its forced positions are all at the top.
Does every puzzle here have exactly one answer?
Yes, and it is checked rather than intended. Each generated puzzle is passed to a solver that counts solutions and stops at two; anything with a second answer is discarded and regenerated. The solver honours all three rules, including connectivity, since a great many bridge arrangements satisfy every number while leaving the map in two pieces.
Why does my solution not count when all the numbers are right?
Almost certainly because the map is in more than one piece. Satisfying every island independently is easy; the connectivity rule is what makes the puzzle a puzzle. The status line says so explicitly rather than just refusing, because "every number is correct and it is still wrong" is baffling the first time it happens.
Can two bridges run between the same pair of islands?
Two, and no more. A double bridge is drawn as a pair of parallel lines and counts two toward each island. Clicking a pair repeatedly cycles through one, two and none, which is quicker than hunting for a separate remove control and matches how most implementations behave.
Can a bridge cross another bridge?
No, and that constraint does much more work than it appears to. It means a long bridge across the middle of the board rules out every crossing bridge at once, which frequently forces a whole region. When a puzzle seems to have no next move, looking for what a long bridge has already ruled out is usually more productive than rechecking the numbers.
Is Hashiwokakero solvable without guessing?
The puzzles here are, by construction — each has a single solution reachable by deduction, and the generator throws away anything ambiguous. In the wild that is not guaranteed: some published puzzles require a trial that only a computer would enjoy. If you find yourself branching on a coin flip here, there is a deduction you have not spotted.
Does this store anything?
Only if you post a time. The puzzle is generated and checked entirely in your browser, nothing is uploaded, and nothing is kept between puzzles unless you choose to put a time on the global board, which sends the name you type and the number.
🔒 This tool runs entirely in your browser. Nothing you enter is uploaded, logged, or stored.