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 apologize for publishing this prematurely; I was in the process of posting a comment clarifying my parenthetical expression (6 going up and 6 going down in a continuous cycle), referring to the 12 segments of each elevator's cycle. I must have pressed the post button for the puzzle instead of the submit for the comment. But I don't remember getting the confirmation panel. Oh well....
The puzzle is not original with me; Paul Nahin, in Digital Dice, says Gamow and Stern actually originated the puzzle. From the questions that have arisen here, I see this puzzle, like many in the puzzle world, ignore some real-life considerations, such as elevators not really going through their full cycle, the way subway trains do in a similar probability puzzle. The solution proposed by the originators assumes something like what are called Sabbath elevators, found mostly in hospitals, where strictly orthodox Jewish patients and doctors need not activate the elevator, but just wait for it to arrive.
I'm sure someone must have noticed that this puzzle would not work very well if its protagonist was on the top or bottom floor, where the elevator always arrives in the "wrong" direction, but actually really is going in the right direction after stopping.
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Posted by Charlie
on 2020-07-27 07:03:52 |
clarifications
