George Gamow and Marvin Stern occupied offices on the second and sixth floors of a seven-story building, and noted that when either took the elevator to the other's floor, it was going the wrong way. It's apparent why: there were ten segments of the elevator's 12-segment cycle (6 going up and 6 going down in a continuous cycle) where the first elevator arrival would be going the wrong way and only two segments where it would be going the desired direction the next time it passed the boarding floor.
But what if a second elevator were placed in the building. What would the probability be that the next elevator to arrive would be going the wrong way? Ignore stops along the way, as they do not affect the distance that need be traveled and probably have more of them for longer trips. The two elevators move independently of each other.
Gamow himself did not get the correct answer for the two-elevator case, but the correct answer was found by Donald Knuth.
I live on the top floor of am 11-story building.
When I want to go down, the elevator will virtually always be going 'the wrong way' unless someone has just got out of it.
Indeed, as it happens I have two elevators, and can confirm the same is true of both.
What is true in the extreme case must also be true to a lesser extent of the penultimate floors.
However, there is a definitional as well as a directional issue here. Say I am on the 11th Floor waiting to go down. The problem only looks for 'segments where (the elevator) would be going the desired direction the next time it passed the boarding floor.'
But this is not the whole story. In reality the elevator is regrettably going 'the right way' all too often when I am waiting for it; say, on the 10th floor going down. But since I am unfortunately not on it, it has not 'passed the boarding floor', so the observation is discarded.
If we reincorporate these cases into the analysis, then I suspect the elevator's behaviour would be much more even-handed.
Edited on July 27, 2020, 5:17 am
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Posted by broll
on 2020-07-27 05:09:53 |
The extreme case
