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In Catan, can player one build a road around a hex, and player two build a settlement between two of player one's road sections while being connected to his own roads from the third side? If so, does this affect Longest Road?

What about the opposite case: if two players have roads coming up to a corner where one of them already has a settlement, can the other one continue building from the other side?

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Yes. You can build the settlement in the middle of an opponent's road (as long as your own road connects to it and there is at least one gap before any other existing settlements).

Yes. It does affect the longest road: the road stops at the settlement for counting purposes, and starts again at the other side.

Yes. Both players who possess roads into an intersection have the option of building the third road out, regardless of whether a settlement is there or whose it is.

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