The Connectivity Rule: How "Everything Must Join" Solves Hashi
Why Hashi (Bridges) forbids closing off a group early, the 1-1 and 2-2 traps, how to count what a group still needs, and a full puzzle solved with these ideas.
The connectivity rule in Hashi (Bridges) says that when the puzzle is finished, every island must be joined into one network, so you can travel from any island to any other along the bridges. That makes it illegal to complete a small group of islands while others are still unconnected. Used the other way round, the rule is a strong deduction tool: any bridge that would seal off a group is ruled out, and that often forces the bridge you need. This article is for players who know the basic rules.
What the connectivity rule says
The connectivity rule is the last of Hashi's rules, and the one with no equivalent in sudoku (more on what does carry over in Bridges for sudoku players). Nikoli, the Japanese publisher that created the puzzle, puts it this way on its rules page: "The bridges will form a continuous link between all the islands." Puzzle Picnic says the same thing in plainer words: every island must end up connected in one group.
The other rules are all local. They talk about one island (its number), one pair (at most two bridges) or one line (no crossings, no diagonals). The connectivity rule is the only one about the whole board. That is why it catches mistakes the others miss, and why it can force a bridge on one side of the board because of islands on the other.
Two terms make the rest of this article easier:
- A group is a set of islands joined to each other by bridges you have already drawn. At the start every island is its own group; at the end there is one.
- A group is closed when every island in it is complete. No new bridge can leave a closed group, so if any island is still outside it, the puzzle can no longer be solved.
The numbers alone are not enough
A grid can satisfy every island's number and still be wrong, because it splits into separate groups. The small puzzle below shows this. It has six islands and exactly one solution, but two ways to make every number right.
Here is the tempting fill. Every island has its number of bridges, and nothing crosses.
The left column (1, 3 and 2) is a closed group, and so is the 1, 2 and 1 on the right. The only correct solution joins them:
The logic to get there takes three steps. The top-left 1 has only one neighbour, the 3 below it, so they share a single bridge. The 3 then needs two more bridges, from the 1 to its right and the 2 below it. The only other way to give it two is a double bridge down to the 2, but that would complete the 1, the 3 and the 2 together and close the group. So the 3 sends one bridge right and one down, and the bottom row follows.
Why closing a group early is never allowed
A closed group can never be reopened, because a complete island cannot take another bridge. If a group closes while any island is still outside it, those outside islands can never be reached, and the puzzle has no solution from that point on.
So the rule for solving is simple: never make a move that closes a group unless that group contains every island on the board. In practice that gives you two kinds of deduction.
- A bridge that would close a group is ruled out. You can mark that link as "no more bridges here".
- Ruling it out forces something else. The island still needs its bridges, so they must go to its other neighbours, and often only one is left.
The Conceptis guide to Hashi techniques treats this as a whole family of isolation techniques, starting with two islands and building up to larger segments. Wikipedia's article on the puzzle describes the same idea as avoiding a "short circuit": a closed subnetwork that cannot connect to the rest.
The two pairs to learn by sight
Two small cases of the connectivity rule come up in almost every Hashi puzzle, and it pays to see them instantly.
- 1 and 1. A single bridge between two 1s completes both. The pair is closed, so two 1s are never joined (unless the whole puzzle is those two islands).
- 2 and 2 with a double bridge. A double bridge between two 2s completes both, with the same result. Two 2s can share a single bridge, but never a double.
Every other pair is safe, because at least one island still needs bridges after the link is made:
| Pair | Link | Safe? | Why |
|---|---|---|---|
| 1 and 1 | single | never | both islands complete, pair closed |
| 2 and 2 | double | never | both islands complete, pair closed |
| 2 and 2 | single | yes | each still needs one more |
| 1 and 2 | single | yes | the 2 still needs one more |
| 2 and 3 | double | yes | the 3 still needs one more |
A pair that is already joined to other islands is a different question, which the next section handles. But at the start of a puzzle, every 1 next to a 1 and every 2 next to a 2 is a gift: it tells you where the bridges do not go.
Counting what a group still needs
For bigger groups, add up what every island in the group still needs. Call that the group's open count. Each new bridge inside the group, or between two groups you are joining, lowers the combined open count by 2: one for each end.
The deduction follows directly. If a move would bring a group's open count to zero while islands remain outside, the move closes the group and is ruled out. Before you add a bridge, take the islands it would join, add up what they still need, subtract 2 for the new bridge, and check that something is left.
A second deduction comes from the same idea, seen from outside. If a group still needs bridges and has only one link left that could lead out of it, that link must be used, because the group has to join the rest somehow. This "last way out" move is less common than the closing check, but on large boards it often breaks a stuck position.
The open count is also the quickest way to check your own work. If you finish a group by accident, the count tells you immediately that something earlier was wrong.
The spare-bridge count: a shortcut from the totals
A puzzle's numbers tell you, before you draw anything, how many double bridges and loops the solution can have. The shortcut follows from the connectivity rule and a little arithmetic.
Every bridge adds 1 to the islands at both of its ends, so the total of all the numbers is exactly twice the number of bridges. A network that joins N islands needs at least N minus 1 linked pairs. Subtract, and you get the spare bridges:
When the spare count is zero, every bridge in the solution is single and the network has no loops at all. The six-island puzzle above is like that: its numbers add up to 10, which is 5 bridges, and 6 islands need at least 5 links. So all five bridges are single, which forces the 3 to send one bridge to each of its three neighbours straight away.
When the count is small, it still helps. Once you have found as many double bridges as there are spares, every remaining link must be single and no loop can form. Some computer versions make the no-loop idea a rule of their own: Simon Tatham's free collection offers an option that forbids loops. In the standard puzzle, loops are allowed, but the count limits them.
A full puzzle solved with connectivity
The puzzle below has eleven islands and one solution. Counting gets you halfway; the connectivity rule does the rest, twice.
Moves 1 to 3: counting first
The 1 on the left edge sees nothing above or below it, so its only neighbour is the 3 to its right: a single bridge. That bridge runs across the second column, so the 2 at the top can no longer reach the 1 below it. The 2's only neighbour is now the 3 to its right, so it takes a double bridge. That 3 then needs one more bridge, and its only other neighbour is the 3 below it: a single bridge down.
The middle 3 of the third row now has two bridges and needs one more, from the 3 on its right or the 3 below. The 3 on the right edge needs three bridges and has just two neighbours: the middle 3, which can give it at most one, and the 4 below it. So the right 3 takes one bridge from the middle 3 and a double bridge from the 4. That completes the middle 3, which sends nothing down.
Move 4: the 1-1 pair
The two 1s in the second column face each other. Joining them would close a pair, so they are never joined. Each takes its single bridge to its only other neighbour: the upper 1 to the 3 on its right, the lower 1 to the 2 on its right.
Move 5: the group of eight
Now the board has four groups: the six islands across the top, the upper 1 with its 3, the lower 1 with its 2, and the 1 in the bottom-right corner on its own.
The 3 in the middle of the fifth row has one bridge and needs two more, from the 4 on its right or the 2 below. The 4 has two bridges and needs two more, from that 3 or the 1 below it. Consider a double bridge between the 3 and the 4. Count the open total of the islands it would join: the 3 still needs 2, the 4 still needs 2, and every other island in those two groups is already complete. That is 4, and a double bridge uses all 4. The top six islands and the 1 and 3 would form a closed group of eight, with three islands still outside. Ruled out.
No bridge at all between them fails too: the 4 would then need both of its bridges from the 1 below, which can take only one. So the 3 and the 4 share a single bridge, and the 3's last bridge goes down to the 2. The spare-bridge count tells the same story: the numbers add up to 24, which is 12 bridges, and 11 islands need at least 10 links, so there are only two spares. The two double bridges already drawn have used both, so every remaining link is single.
Move 6: the corner
The bottom 2 is now complete, with one bridge from the 1 and one from the 3. The 4 has three bridges and needs one more, and its only open neighbour is the 1 below it. That single bridge completes the puzzle and joins all eleven islands. The 1 in the corner was the last island outside the network, one of the three that the group of eight would have shut out.
Frequently asked questions
Can two 1s ever be connected in Hashi?
Only if the puzzle has exactly two islands. In any real puzzle, a bridge between two 1s completes both and leaves them cut off, which breaks the rule that all islands must join one network.
Are loops allowed in Hashi?
Yes, in the standard rules. A route of bridges can go round in a ring and come back to where it started. Some computer versions offer a no-loops option, but Nikoli's rules do not forbid loops, and neither do the rules Puzzle Picnic teaches.
Why does my grid look finished but not count as solved?
Check whether every island is joined into one network. It is possible to satisfy every number and still split the board into two or more separate groups. Trace a route from one island to all the others; if one is unreachable, a closed group formed somewhere.
Is the connectivity rule needed on easy puzzles?
Often, yes. Even small puzzles tend to include a 1 next to a 1 or a 2 next to a 2, and avoiding those closed pairs is frequently the move that breaks the puzzle open.
Where to go next
The connectivity rule is one of six core ideas in the Hashi strategies guide, which shows how it combines with counting and crossing blocks. If closed groups keep catching you out, the common Bridges mistakes include a quick way to find where a group went wrong. New to the puzzle? Start with how to play Bridges, or play Bridges and look for the 1-1 pairs on your first board.