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Solutions to the Gamow-Stern Elevator Problem, a(n) = ceiling((log_10 5)/(log_10 (1+2(n-2)))) for integer n >= 3.
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%I #16 Jan 06 2017 13:24:39

%S 2,3,4,4,5,6,7,8,9,9,10,11,12,13,13,14,15,16,17,17,18,19,20,21,21,22,

%T 23,24,25,25,26,27,28,29,29,30,31,32,33,33,34,35,36,37,38,38,39,40,41,

%U 42,42,43,44,45,46,47,48,49,50,50,51,52,53,54,54,55,56,57

%N Solutions to the Gamow-Stern Elevator Problem, a(n) = ceiling((log_10 5)/(log_10 (1+2(n-2)))) for integer n >= 3.

%C Gives the number a(n) of elevators in an ideal system needed for a building with n floors to have a probability greater than or equal to 0.4 that the first elevator arriving to a given floor will be on its way down.

%C Also the least positive integer n such that 1/2 + (1/2)(1-2p)|1-2p|^(n-1) >= 0.4.

%C 1/2 + (1/2)(1-2p)|1-2p|^(n-1) approaches 1/2 as n goes to infinity.

%D George Gamow and Marvin Stern, Puzzle-Math, (New York: Viking Press, 1958).

%D Donald E. Knuth, Fundamental Algorithms, Volume 1 of The Art of Computer Programming (Reading, Massachusetts: Addison-Wesley, 1968).

%D Donald E. Knuth, Selected Papers on Fun and Games, (Stanford, California: CSLI Publications, 2011), pages 79-86.

%F a(n) = ceiling((log_10 5)/(log_10 (1+2/(n-2)))) for integer n >= 3.

%e For n=20, a(20) = ceiling((log_10 5)/(log_10 (1+2/(20-2))))= ceiling((log_10 5)/(log_10 (10/9))) = 16.

%t a[n_] := Print[Ceiling[(Log10[5])/(Log10[1+2/(n-2)])]]

%t For[i = 3, True, i++, a[i]]

%K easy,nonn

%O 3,1

%A _Luke Botta_, Jan 03 2017