Is the Elevator Puzzle a Math Paradox or a Paranoid Delusion?

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The Simplified General Single-Elevator Puzzle

Other Elevator Puzzles with 10 Floors. Image by Mike DeHaan

(Use this 10-floor image to help think about other situations.)

Rather than 10 floors, there are ‘n’ floors in a building with a single elevator that always stops at every floor. To make matters simpler, let’s just think about a single trip at any one time.

The elevator has “2*n – 2” possible states. On the first floor, it can only be in the “1 : up” state. On the top floor, the only state is “n: down”. On every other ith floor, the elevator has two states: “i : up” or “i : down”.

Let us also stipulate “n > 1”. (A building with only one floor has no need of an elevator. A building with zero or fewer floor levels has trouble existing.)

Start Up from the First Floor

When you first enter the building, you are on the first floor and can only go up. Since the first floor is also the lowest level for the elevator, you will never be disappointed by seeing the elevator door open but indicate it will go down.

The elevator scores 100% satisfaction when you start from either extreme, because it can only go in the direction you want to go. (On the other hand, it is almost certain you have to wait for it to travel).

Based on the Second Floor

If the elevator is on the first floor, or just arriving at the second floor and heading up, you will be pleasantly surprised only two times out of the “2*n – 2” states. That of course means you will see the elevator in state “2 : down” in “(2*n – 4)” cases out of “(2*n – 2)”.

However, if you are on the second floor but want to go down, your probability of seeing the “2 : down” elevator is “(2*n – 4)/(2*n – 2)”.

The Plot Thickens on the Third Floor

The for good states for the elevator are “2: down”, “1 : up”, “2 : up” and “3 : up” if you want to go higher. Therefore the probability of success is “4/(2*n – 2)”. Going down, however, you would succeed “(2*n – 6)/(2*n – 2)”% of the time.

The Generalized Elevator Puzzle for the ‘i’th Floor

We stipulate that “n > i > 1” for this discussion, where ‘n’ is the number of floors and ‘i’ is the floor where you are currently. In the general case, you are on floor ‘i’, so there are “i – 1” floors below and “n – i” floors above.

The elevator has “2n – 2” possible states, pairing a floor and a direction. These states are {(1 : up), (2 : up),…,(n-1 : up), (n : down), (n-1 : down), (n-2 : down),…,(3 : down), (2 : down)}.

We want the “happy” state where elevator is heading in the direction you want, rather than going the other way.

Trivial Cases of the Elevator Puzzle

Let’s eliminate the trivial cases. When you are on the first floor, your only desire is to go up, and the elevator will never head down from the first floor. Likewise, on the nth floor, you can only want to go down, and the elevator will never go up from there. In both cases, you have a 100% guarantee of happiness.

As discussed in the ten-story building, when you start one floor away from the top or bottom, there are only two “Happy!” states, out of “2n – 2” possibilities, if you are going toward the center of the building.

Generally, Things are Looking Up for the Elevator Puzzle

This section deals with going up.

In the general case, you are on floor ‘i’, so there are “i – 1” floors below and “n – i” floors above. We restrict ‘i’ by stating “1 < i < (n-1)”.

When you want to go up, you would become “happy” in three situations as you step into the lobby. You hit the jackpot if the elevator immediately stops on your floor indicating “up”: this was the unique state “(i : up)”. If the elevator was in any state {(1 : up), (2 : up),…,(i-1 : up)}, you had to wait but never saw the elevator going down. Also, if the elevator was in state {(i-1 : down),…,(2 : down)}, you had a longer wait but, again, never saw the elevator on your floor but going down.

So the number of “happy” states when you head up from the ith floor are “1 + (i – 1) + (i – 2) = 2*i – 2”. The probability of being happy is “(2*i – 2)/(2*n – 2)”.

Therefore, when you want to go up, the probability that the elevator will first be on its way up is lower as you start on lower floors… with the exception of starting on the first floor.

Down is Just the Reverse of Up

Finally, note that the situations are the same as you head down from the top floors. The top floor will always be seen in the desirable “n : down” state. The (n-1)th floor has two desirable states when you want to go down, but “2*n – 2” desirable for going up.

The “happy” states are: (i : down); {(n : down), (n-1, down),…,(i+1, down)}; and {(i+1, up), (i+2, up),…,(n-1, up)}. There are “1 + (n – i – 1) + (n – i – 2) = 2*n – 2*i – 2” happy states out of “(2*n – 2)”.

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