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This topic contains two sections:
The other most basic hashi technique is the Partial Combinations technique. The rule for this technique is If an island has N neighbours that it can bridge to and requires 2N-1 links, then it must bridge at least once to all its neighbours. Here is an example:
In the above example, the orange '3' must bridge to the red '3's above and below it, as shown. To see why, consider that we must have 3 bridges leaving the orange '3'. We can have at most two bridges in each direction. So if we put two bridges to the island above, we still need one bridge to the island below, and even if we put two bridges to the island below, we still need one to the island above. So the orange island must bridge at least once to the islands above and below it, as shown in blue. Here is another example:
In this case, the '6' highlighted in orange must bridge at least once to both the '4' and the '5' highlighted in red, as shown. Looking at the '6', we see that it already has 3 bridges, linking it to the islands above and to the right. Both of those islands have been completed, meaning that the remaining 3 bridges must be shared between the islands below and to the left. No matter how we share these 3 bridges between the two red islands, both of them will be bridged to at least once, so we can add one bridge to each, as shown in blue.
We can extend the Partial Combinations technique to cover some extra cases which are not immediately obvious. The rule for extended Partials is If an island has N neighbours that it can bridge to twice, M neighbours that it can only bridge to once, and 2N + M - 1 bridges remaining, then it must bridge to all the islands that it can bridge to twice. This may seem very complicated, but its actually quite easy to spot and use:
The '3' highlighted in orange must bridge at least once to the '2' highlighted in red, as shown. This is because it can bridge at most once to the two '1' islands to the left and below, so no matter what happens, there is at least one bridge left over which must go to the red '2'. Here is a more complex example:
In the above example the '5' highlighted in orange must bridge to the two '3' islands highlighted in red, as shown. The orange island has 4 bridges left, and can still bridge to three islands: the two highlighted in red and also the '2' island to the right. The '2' island to the right, however, has only one bridge remaining. So the orange island would at most be able to link there once. Thus at least 3 bridges must be shared between the two islands highlighted in red. This means, as before, that both of the red '3's must be bridged to at least once, as shown.
Copyright © Adam A. Brown, 2006. www.sudokutiger.com
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