Each lock has a room number, and a list of random numbers stored in the little computer in the lock. It keeps track of which number from the list is “active”.
The computer at reception has a list of numbers for each room’s lock, and also which one is “active”. They use the computer to store a room number and key number on the card.
When they create a new key at the front desk, they make it with the next number on the lock’s list. When you try to open the door with it, the lock gets the room and key number and thinks, “wait a second, let me check the list… that’s not the current key number… but it is the next one on the list, so I guess this is the next guest.” The lock makes the next number on the list “active” and that key will continue to work until a new key comes along with the next number on the list. If you try a key with the wrong number, or a key number that isn’t the “active” one or the next on on the list, the lock simply doesn’t open.
Some systems also include a date, so that a card will automatically stop working after a certain amount of time.
There’s variations on the theme, but they work similarly, so no radios or extra wiring is necessary. They also have special numbers for hotel staff “master keys” which are handled differently. Places with stricter security can get locks that have wired (usually) connections to computer systems that use more complicated ways determining who the card holder is and what they have permission to access.
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