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Is the following setup allowed for the red player to get from Essen to Stockholm?

The theory being the red player is using the green players route to get from Essen to Copenhagen and then again from Copenhagen to Stockholm. enter image description here

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Yes.

According to the rules:

Building a train station

A Train Station allows its owner to use one, and only one, of the routes belonging to another player, into (or out of) that city to help him connect the cities on his Destination Tickets.

Stations may be built on any unoccupied city, even if it currently has no claimed routes into it. Two players may never build a Station in the same city.

Each player may build a maximum of one Station per turn, and three Stations throughout the course of the game.

To build his first Station, a player plays and discards one Train card from his hand, and places one of his colored Train Stations on the chosen city.To build a second Station, the player must play and discard a set of two cards of any one color; and to build his third, a set of three Train cards of any one color. As usual, you can replace any number of cards by Locomotives.

If a player uses the same Station to help connect cities on several different Tickets, he must use the same route into the city with the Station for all of those Tickets. The Train Station owner does not need to decide which route he will use until the end of the game.

A player is never required to build any Stations. For each Station a player has not used, four points are added to his score at the end of the game.

So as long as:

  1. you built at most one station per round.
  2. nobody else build a station on that city.
  3. you have enough stations left.
  4. you can pay for the stations.
  5. you only use each station for one connection between two cities.

This is certainly possible. in this case, you are free to go.

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    As I recall, the station figures are shaped so that they fit on top of the trains. So, while the rulebook doesn't say it, it hints at the idea of moving the station on top of the "borrowed" route to mark the choice in the end. The station can only ever borrow one route, so that's ok. (Just the choice of which route is delayed to the end of the game.) And with the station marking the route, it doesn't matter which end of the route the station came from.
    – ilkkachu
    Sep 3 at 11:49

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