Effy grabs passport, but doesn't use it?
When William and Effy first show up to the hotel William is staying at, they ask for Effy's passport. Effy doesn't have it so they go to Effy's house, where we see Effy grab her passport. When they return the hotel, William sneaks Effy in through the back door, thus circumventing the need for the passport.
What was the purpose of Effy grabbing the passport? I doubt she intended to run away with William; she grabbed clothes but those were presumably to change into/travel to William's grandfather, not leave the country.
That whole sequence seemed really weird.
Is it really the law/common practice in Copenhagen/Denmark to require passports to keep on file when checking into hotels? If it is, the only answer I can think of is William snuck her in to avoid any paper trail with her (but why didn't he do that the first time)?
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When logic and science aren't on your side, you always lose.