Plankway

At least one: bridges you know exist before you know how many

Sometimes you know that a bridge is there without knowing yet whether it is single or double. That is still valuable: it fixes the connection, and it often finishes the island on the other end.

The idea

Pick one connection of an island. Add up the most the other connections could give. If that is less than the island’s number, the difference has to come from the connection you picked.

ABCD1234322
A 3 with two neighbours: each side can give at most 2, so each side must give at least 1.

The well-known cases: a 3 with two neighbours, a 5 with three neighbours and a 7 with four. In each case every neighbour gets at least one bridge.

With a small neighbour

Small neighbours lower the maximum. Below, the 4 in C2 has three neighbours, but one of them is a 1 that can give only one bridge. The most it could get is 1 + 2 + 2 = 5, just one more than it needs. So leaving out the right or the lower connection would leave at most 3, not enough: both need at least one bridge.

ABCDE12344132
The 4 in C2 needs at least one bridge to the right and one down; the 1 on the left is still open.

How it fits with the rest

At least one is a softer version of every bridge counts: that one applies when the maximum exactly matches the number, this one when it is slightly more. In Plankway it is the first technique that is not purely "count and fill", which is why Monday puzzles hardly need it and Tuesday puzzles start to. After placing the single bridge, check its other end: with a 1 or a 2 there, that island may now be full.

Recent puzzles that needed this technique

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