Question: How did Bond know which room Goldfinger had at the hotel?
Answer: He deduced that Goldfinger was using a partner to spy on his opponent's hand, and to check his theory he went to the room with the best line of sight. Alternatively, he went (off-screen) to the desk and used his charm, which was utterly irresistible in the Bond films of the '60s, to find out where Goldfinger was staying.
Answer: Security and guest privacy was less of a concern in this era. Often someone could merely inquire at the desk which room a guest was staying in. Another ploy often used in movies was to leave a note for the guest and then watch which numbered mailbox the concierge placed it in.
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