Hashi Strategies: The Deductions That Solve Most Bridges Puzzles
The Hashi (Bridges) techniques in the order you need them: corner 4s, lonely 1s, crossing blocks, closed groups and what-if reasoning, with worked grids.
The deductions that solve most Hashi (Bridges) puzzles all compare two numbers: how many bridges an island still needs, and how many its neighbours can still take. When they are equal, every bridge is forced. When the island is one short, every neighbour gets at least one. Add three more ideas (one neighbour left, crossing blocks and never closing a group early) and you can solve nearly every Easy and Medium puzzle without guessing. This guide is for players who know the rules and want a method.
If you are new to the puzzle, read how to play Bridges first. The techniques below follow the order you will meet them, and a full puzzle near the end uses them together.
Read the board first: neighbours and capacity
Every Hashi technique starts with two facts about an island: its neighbours and its capacity. A neighbour is the first island in each of the four directions, as long as no bridge already blocks the way. Capacity is the most bridges a neighbour can still take from this island.
A neighbour's capacity is the smallest of three things:
- 2, because no pair of islands can have more than two bridges.
- What the neighbour still needs. A 1 can take only one bridge. A 3 that already has two bridges can take only one more.
- Nothing, if a bridge is in the way. A bridge that crosses the line between the two islands cuts them off for good.
Add up the capacities of an island's neighbours and you have its maximum. Compare that with what the island still needs, which is its number minus the bridges it already has. Almost every technique in this guide is a version of that comparison.
A useful way to keep track while you solve: think of each island by what it still needs, not by its printed number. A 5 with three bridges is a 2 for the rest of the puzzle. A 3 whose neighbour has just been completed has one fewer neighbour. The printed number tells you where an island started; the "still needs" number tells you where the next move is.
The fully forced islands
Two techniques decide an island completely: every bridge it can take is forced, or it has only one place left to send them. Look for these first on every board.
Technique 1: islands that must use every bridge
When an island's neighbours can take exactly the number of bridges it needs, every possible bridge is forced. The classic cases are islands whose number is twice their neighbour count: a 4 with two neighbours, a 6 with three and an 8 with four. Each must have a double bridge to every neighbour.
What matters is the number of neighbours, not where the island sits. An island in a corner usually has two, and one on an edge usually has three, so these are good places to look first. But an island in the middle of the board with islands only above and to the left behaves exactly like a corner island, and a 4 there is just as forced.
| Neighbours | Forced when the number is | What you draw |
|---|---|---|
| 1 | 1 or 2 | everything to that one neighbour |
| 2 | 4 | a double bridge to both |
| 3 | 6 | a double bridge to all three |
| 4 | 8 | a double bridge to all four |
An 8 is the extreme case: it always has four neighbours and always takes two bridges to each. The Conceptis guide to Hashi techniques lists the corner 4, the edge 6 and the middle 8 as its starting techniques, the moves to make before anything else.
Technique 2: one neighbour left
An island with only one neighbour it can still reach must send all its remaining bridges there. At the start of a puzzle this applies to a 1 or a 2 that sees only one other island. Later in the solve it is the most common move of all, because as islands fill up and bridges block lines of sight, more and more islands are left with a single option.
The rule needs a check: the remaining bridges must fit. A 2 with one neighbour needs a double bridge, which works only if that neighbour still needs at least two. If it does not, something earlier has gone wrong.
This is the technique that keeps a solve moving. Each forced bridge completes an island or blocks a line, and that often leaves another island with one neighbour. Whenever you draw a bridge, look at the islands around it and ask whether any of them just lost its last alternative.
The partly forced islands
The next two techniques do not finish an island, but they give you certain single bridges, and those singles block crossings and shrink what the neighbours need.
Technique 3: islands one short of full
When an island needs one bridge fewer than its neighbours can take, it must send at least one bridge to every neighbour that can take two. The classic cases are odd numbers: a 3 with two neighbours, a 5 with three and a 7 with four.
The reason is simple counting. Take a 3 in a corner with two neighbours that can each take two. If one neighbour got no bridge, the other would have to take all three, and no pair can have three bridges. So each neighbour gets at least one. The third bridge goes to one of them, but you do not know which yet.
| Neighbours | Number one short of full | What you know |
|---|---|---|
| 2 | 3 | at least one bridge to each |
| 3 | 5 | at least one bridge to each |
| 4 | 7 | at least one bridge to each |
Draw the single bridges straight away. They are certain, and each one may block a crossing bridge elsewhere.
The general form of this technique works for any number. To find the fewest bridges an island must send to one neighbour, subtract the combined capacity of all its other neighbours from what it still needs. If the answer is 1 or 2, draw that many bridges. A 3 with two neighbours of capacity two: 3 minus 2 leaves 1 for each. An 8 with four neighbours: 8 minus 6 leaves 2 for each, which is Technique 1 again.
Technique 4: small neighbours change the sums
A neighbour that can take only one bridge, such as a 1 or a partly finished island, lowers an island's maximum, and that turns many unclear islands into forced ones. Watch for 1s next to big numbers.
Take a 5 on the top edge with three neighbours: a 1, a 3 and a 4. Its maximum is 1 + 2 + 2 = 5, exactly its number, so the 5 takes everything: a single bridge to the 1 and a double to each of the others. Without the 1 next to it, the same 5 would only be one short of full (Technique 3).
The same pattern turns up in other shapes:
| Island | Neighbours | Maximum | Result |
|---|---|---|---|
| 3 | a 1 and one other | 1 + 2 = 3 | single to the 1, double to the other |
| 5 | a 1 and two others | 1 + 2 + 2 = 5 | single to the 1, doubles to the others |
| 7 | a 1 and three others | 1 + 2 + 2 + 2 = 7 | single to the 1, doubles to the others |
| 4 | a 1 and two others | 1 + 2 + 2 = 5 | at least one bridge to each of the two others |
| 6 | a 1 and three others | 1 + 2 + 2 + 2 = 7 | at least one bridge to each of the three others |
The last two rows are Technique 3 in disguise: once the 1 lowers the maximum, the island is one short of full, so every neighbour that can take two gets at least one. The 1 itself gets nothing certain yet.
Partly finished neighbours work the same way. A 2 that already has one bridge elsewhere has capacity one, just like a 1. Recalculate the sums each time a neighbour gains a bridge.
Technique 5: crossing blocks
Every bridge you draw blocks any bridge that would cross it, and that removes a neighbour from up to two other islands. Beginners often miss it, because it happens away from the island they were just looking at.
After drawing a long bridge, especially a vertical one through the middle of the board, run your eye along it and look for islands on either side that could have reached each other across it. They no longer can. Each of them has one fewer neighbour, which may leave it with only one (Technique 2) or with neighbours that now exactly match its number (Technique 1).
The full puzzle below shows the effect: a double bridge in the middle column cuts a row in two, and two islands on that row are forced at once. Crossing blocks also work in reverse as a warning: before you draw a bridge, check that it does not cross one you have already drawn. A crossing is never allowed, not even briefly.
Technique 6: never close a group early
The final rule of Hashi says every island must join one network. So any set of moves that would finish a small group of islands, with no bridge left to leave it, is illegal while other islands are still outside. This is the isolation technique, and it is where Hashi differs most from number puzzles like sudoku.
Two cases are worth learning by sight:
- Two 1s are never joined. A single bridge between them would complete both islands and leave the pair stranded. So each 1 must use its other neighbours.
- Two 2s never share a double bridge. For the same reason: both would be complete and cut off together. A single bridge between two 2s is fine.
The same reasoning works for bigger groups. Add up what every island in a joined group still needs. If one more bridge inside the group, or one bridge to a neighbour, would bring that total to zero while islands remain outside, that bridge cannot be drawn. And if a group still needs bridges but has only one connection left to the outside, that connection must be used.
Isolation deserves more space than a section, so it has its own article: the connectivity rule in Hashi, with full puzzles where it is the only way forward. Closing a pair by accident also appears in our list of Bridges mistakes.
A full puzzle, solved technique by technique
The puzzle below has eleven islands on a 7 by 6 board and one solution. Every move uses one of the techniques above, and each move is named.
Moves 1 and 2: one neighbour, then a full island
The 1 in the top-right corner sees nothing to its left, so its only neighbour is the 2 below it: a single bridge (Technique 2). The 1 in the second row sees nothing above, below or to its left, so its single bridge goes right to the 3.
That 3 now needs two more bridges, and it has only one other neighbour: the 5 far below it. Nothing sits above the 3 or to its right. So the 3 sends a double bridge down to the 5 (Technique 2 again, or Technique 4 if you counted from the start: a 1 and one other neighbour make 1 + 2 = 3).
Move 3: a crossing block
The new double bridge runs straight through the third row. Before it, the 2 on the left edge and the 2 on the right edge could see each other along that row. Now they cannot (Technique 5).
The left 2 has one neighbour left, the 4 below it, and needs two bridges: a double bridge down. The right 2 already has one bridge from the 1 above it and needs one more; its only other neighbour is the 4 below it, so a single bridge goes down.
Move 4: what the middle row still needs
Now count what is left. The 5 has two bridges and needs three more, from the 4 on its left and the 4 on its right. Two neighbours that can each take two, and it needs three: at least one bridge to each (Technique 3).
The right-hand 4 has one bridge and needs three more, from the 5 and the 2 in the bottom-right corner. Again at least one to each. The left-hand 4 has two bridges and needs two more, from the 5 and the 2 in the bottom-left corner. It is already sure of one bridge to the 5, so at most one can go down to the bottom-left 2.
Move 5: two 2s that must not close
The bottom-left 2 has two neighbours: the 4 above it, which can now give it at most one bridge, and the 2 to its right. A double bridge between the two bottom 2s would complete both and strand them (Technique 6), so they share at most one bridge. The bottom-left 2 therefore takes exactly one bridge from each neighbour.
Move 6: everything falls into place
The left 4 has three bridges and needs one more: it can only go to the 5, so a single bridge goes right. The 5 now has three bridges and needs two more, all from the right-hand 4: a double. The right-hand 4 then has three bridges and needs one: a single bridge down to the corner 2. That 2 needs one more, from the bottom-middle 2, which completes the puzzle.
The whole solve used Techniques 2 to 6 and no guesses. That is typical: on smaller boards these ideas, applied in a loop, are all you need.
When the simple moves run out: what-if reasoning
On bigger and harder boards you will sometimes reach a position where no island is forced by counting alone. The next step is what-if reasoning: pick one undecided bridge, imagine each possible value in turn, follow the forced moves, and see which one breaks a rule. If every option but one leads to a broken rule, the remaining option is proved. This is not guessing, because you never commit to a move you have not proved.
A small puzzle shows the pattern. After the forced moves, four islands are left in a square: two 3s and two 2s, each still needing two bridges, linked in a ring.
Nothing is forced by counting: each of the four islands needs two bridges and has two neighbours in the ring. So ask what the bridge between the two 3s could be.
- None. Then the right 3 needs both of its remaining bridges from the 2 below it, a double. That 2 is complete, and so is the right 3, and so is the 1 in the top-right corner above it. Those three islands form a closed group, cut off from the rest. Not allowed.
- Two. Then both 3s are complete, so each 2 can only get its bridges from the other 2: a double between two 2s, a closed pair. Not allowed.
- One. The only option left. Each 3 then sends one bridge down, and the two 2s share a single bridge.
Keep what-if chains short. One or two forced steps is easy to hold in your head; a long chain is where mistakes creep in. If a chain gets long, try a different bridge first.
Hashi is hard for computers in general: in 2009 Daniel Andersson proved that deciding whether a Hashi puzzle has a solution is NP-complete, which means no fast method is known that works for every possible board. Published puzzles of a normal size are a different matter. They are made to be solved by a chain of reasoning, and when counting runs out, a short what-if like the one above is the usual next step.
How the techniques fit together
A good solving routine runs the cheapest checks first and repeats them after every new bridge. This is the order most solvers settle on.
- Sweep for one-neighbour islands and full islands (Techniques 1 and 2). These moves are instant and certain.
- Draw the sure single bridges from islands one short of full (Technique 3), including the cases made by small neighbours (Technique 4).
- After each new bridge, check what it blocks (Technique 5) and what it completed. Completed islands drop out as neighbours.
- Scan for pairs and small groups that are close to closing (Technique 6).
- Only then, try a what-if on the most constrained undecided bridge.
The computer science literature describes similar sets of rules. A 2012 paper by Reza Firsandaya Malik and colleagues, on solving Hashi with human-style techniques and depth-first search, names five: Just Enough Neighbor, One Unsolved Neighbor, Few Neighbor, Leftovers and Isolation, with a search step for whatever the rules cannot decide. That is the same shape as the routine above: logic first, careful case-checking last.
In Puzzle Picnic, the board grows from 7 x 7 on Easy to 11 x 14 on Expert, and every puzzle has exactly one solution that logic will find. On the bigger boards, the routine matters more than any single trick, because forced moves are spread out and easy to overlook.
Frequently asked questions
What is the best first move in a Hashi puzzle?
Look for islands with no choice: a 1 or 2 with a single neighbour, or an island whose neighbours can take exactly its number, such as a 4 with two neighbours, a 6 with three or an 8. Corners and edges are the usual places to find them.
Does a big number in the middle mean double bridges everywhere?
Only for an 8, or for a number that equals twice its neighbour count. A 7 with four neighbours is one short of full, so it has at least one bridge to each neighbour, but which one gets a single bridge is decided later.
How do I avoid isolating islands?
Before you complete an island, ask whether it and the islands joined to it still have a way out to the rest of the board. Never join two 1s, never put a double bridge between two 2s, and count what a small group still needs before you add its last bridge.
Is it cheating to try a bridge and see what happens?
Testing an idea in your head and following it to a contradiction is standard logic, the same as in sudoku. Drawing a bridge you have not proved and hoping it works out is guessing, and in a well-made puzzle it is never needed.
Do these techniques work on every board size?
Yes. Every technique here looks only at one island, its neighbours or one small group, so it works the same on a 7 x 7 board as on an 11 x 14 one. Bigger boards simply have more islands to check, which is why a fixed routine helps.
Where to go next
Practise the first four techniques on small boards until the corner 4s and lonely 1s jump out at you, then move up a size. The connectivity rule is the next thing to master, and the story of Hashiwokakero tells you where the puzzle came from. To try the techniques on a fresh board, play Bridges in your browser.
Sources
- Nikoli: Hashiwokakero rules
- Conceptis Puzzles: Hashi techniques
- Bulletin of Electrical Engineering and Informatics: Solving Hashiwokakero Puzzle Game with Hashi Solving Techniques and Depth First Search (Malik, Efendi and Pratiwi, 2012)
- Information Processing Letters: Hashiwokakero is NP-complete (Andersson, 2009)
- Wikipedia: Hashiwokakero