Infinity Hotel Paradox
Date: 09/15/1999 at 12:29:23 From: Mr. Tuesday Evening Subject: Infinity Paradox Our math class has a paradox we can't figure out. Can you make heads or tails of it? A hotel has an infinite number of rooms. Each room has already been assigned to a person with the corresponding room number on his or her shirt; i.e. the person with a shirt that says 438 is in room 438, etc. Now an infinite number of buses come, each containing an infinite number of people. They all want a room. Explain how all of these can be accommodated while still maintaining the original premise that each person gets his or her own room. Also, you can only use positive integers, no negatives or irrational numbers. Thanks, -MTE
Date: 09/15/1999 at 13:45:05 From: Doctor Rob Subject: Re: Infinity Pardox Thanks for writing to Ask Dr. Math. One way to do this is as follows. Let the original occupants of the hotel be considered "Bus Zero." Denote the people by an ordered pair, (B,N), where B is the bus number and N is the number of the person on that bus. Arrange the pairs in a rectangular grid: : : : : (0,4) (1,4) (2,4) (3,4) ... (0,3) (1,3) (2,3) (3,3) ... (0,2) (1,2) (2,2) (3,2) ... (0,1) (1,1) (2,1) (3,1) ... Now take the person in the diagonal containing (0,1) only, and give him the first room. Then take the people in the diagonal from (0,2) to (1,1), in order, and give them the next two rooms. Then take the people in the diagonal from (0,3) to (2,1), in order, and give them the next three rooms. Then take the people in the diagonal from (0,4) to (3,1), in order, and give them the next four rooms. Then take the people in the diagonal from (0,5) to (4,1), in order, and give them the next five rooms. Continue in this way forever. Now every person in every bus will have a room. Person N in bus B will have been taken in the diagonal from (0,B+N) to (B+N-1,1). In fact, he or she will have been given the following room number R: R = (B+N)*(B+N+1)/2 - N + 1. No two people will have the same room. (I leave that to you to verify.) Room R will be on the D-th diagonal, where D = (1+sqrt[8*R])/2, rounded down to the nearest integer, and it will be assigned to the N-th person on Bus B, where B = R - D*(D-1)/2 - 1, N = D*(D+1)/2 - R + 1. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/
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