I couldn't help but remember this urban legend about the elevator game. It's spooky, but at the same time incites you to try doing it. I think just the rule of needing to go to some random combination of floors in anticipation that something will happen, does the job.
This principle could be used to design games to entertain people when the elevator is less used (say in the late evening or at night or at any time of the day if the building is quiet) and again - take money for it. So basically this would be a hybrid between a slot machine and an elevator - put some coins in and you can play.
You put the coins in, press "play" and the screen starts displaying you the instructions, e.g. "press the 2nd-floor button" - once you got there and the door closed after opening, it continues: "press the 3rd-floor button", then "press the 5th.." after you got to the 5th floor it asks you to guess the next floor you need to go to. In this specific case, the idea behind the combination of numbers is that 2,3, and 5 are all prime numbers, so the next one would be seven.
If you guessed that right you'd get some prize, it could be giving you your coins back or doubling them. But it would be more interesting if something happened outside of the elevator, on the final floor you got to, for example, the lights of that floor could start blinking, some music could start playing, etc.
The example with prime numbers is rather easy, there could be various mathematical sequences (functions) that are way more difficult to guess, the rules could require you to guess not just one but a few numbers in the sequence, to win the game. Each time you guessed the number right something would happen on that floor after the door opened.
The combination of floors could also code the melody representing notes on a piano keyboard. Once you've finished riding the combination the elevator would play you the melody.
Riding the elevator to all the floors is arguably more interesting than turning handles on a slot machine since it involves you physically moving somewhere. It's like playing chess with real people instead of chess figures.
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