Hashi puzzle tips: six patterns to spot at a glance

Most of a bridges (Hashiwokakero) puzzle is a handful of patterns you can recognise without thinking. Here are six, each with a picture, a lookup table for the counting rule behind them, and a small calculator. Every statement below is a deduction that holds on any valid board.

Pattern 1: the number fills every exit

A 4 in a corner sees two islands, and two islands can take at most two bridges each. 2 + 2 = 4, so both links are doubles. The same holds for a 6 with three neighbours and an 8 anywhere.

Pattern 2: one short of full means at least one each

A 7 with four neighbours: if any link were empty, the other three could carry at most 6. So every link has at least one bridge. A 3 in a corner (two neighbours) and a 5 with three neighbours work the same way.

Pattern 3: a small neighbour caps the link

A link can never hold more bridges than the smaller of its two numbers, and never more than 2. A 1 next to anything takes at most one bridge, so when you count a big island's exits, treat links to 1s as worth 1, not 2. This often tips the counting rule in your favour.

Pattern 4: two 1s must not touch

If two 1s joined, each would be full and the pair would be sealed off from every other island. A finished board is one connected network, so that link (dashed) is empty. Likewise two 2s must not be joined by a double bridge.

Pattern 5: an island with one exit is forced

If the other links from an island are blocked (by a crossing bridge, or by a neighbour that is already full), whatever is left must carry the whole number. A 2 with one open exit gets a double bridge there.

Pattern 6: a certain bridge blocks its crossing

Bridges never cross. Once a bridge is certain, the link that would cross it (dashed here) is empty, and both islands on that link have one fewer exit. Re-run the counting after every certain bridge: it is how one deduction unlocks the next.

The counting rule as a table

For an island with k neighbours it can see, every link takes 0, 1 or 2 bridges. The most the other k − 1 links can hold is 2 × (k − 1), so each link must hold at least number − 2 × (k − 1). That gives this table of bridges you can place for certain on every link (counting every neighbour as able to take a double):

Certain bridges per link, by island number and visible neighbours
Number	1 neighbour	2 neighbours	3 neighbours	4 neighbours
1	1 each	none	none	none
2	2 (full double)	none	none	none
3	impossible	1 each	none	none
4	impossible	2 (full double)	none	none
5	impossible	impossible	1 each	none
6	impossible	impossible	2 (full double)	none
7	impossible	impossible	impossible	1 each
8	impossible	impossible	impossible	2 (full double)

Remember to lower the neighbour count (or the cap on a link) when you have already ruled something out: a neighbour that is full, a 1 (cap of one bridge), or a link that a certain bridge crosses.

Try the rule: bridge budget calculator

Number on the island Neighbours it can still see (open links) Of those, links to a 1 (they take at most one bridge)

Use it on a real board

The daily Causeway puzzle is built so that every board can be finished with these patterns plus the loop rule, never by guessing. Stuck? Paste the board into the bridges solver to see which rule fires next, or read the step-by-step guide.