IBM Research | Ponder This | October 2008 challenges

Ponder This

October 2008

<<September October November>>

Ponder This Challenge:

Consider a frog starting from 0 and jumping from integer to integer. With probability p he makes a jump of +1, with probability (1-p) a jump of -1. Assume .5 < p < 1. Assume each jump is independent. Let B(n,k) denote the binomial coefficient n choose k.

1.

a. How fast will the frog move towards +infinity?

b. How many times on average will the frog visit each non-negative integer?

2. What is the probability the frog will be at 0 after 2*n jumps?
3. Combine 1b and 2 to find the sum from 0 to infinity of (x**n)*B(2*n,n). (0 < x < .25).

We will post the names of those who submit a correct, original solution! If you don't want your name posted then please include such a statement in your submission!

We invite visitors to our website to submit an elegant solution. Send your submission to the ponder@il.ibm.com.

If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Challenge: 10/02/2008 @ 09:00 AM EST
Solution: 11/03/2008 @ 10:00 AM EST
List Updated: 11/06/2008 @ 08:30 AM EST

Dan Dima (10.02.2008 @06:23 PM EDT)
V Balakrishnan (10.02.2008 @08:57 PM EDT)
Alan Murray (10.02.2008 @09:51 PM EDT)
Henry Bottomley (10.02.2008 @09:55 PM EDT)
John T. Robinson (10.02.2008 @11:13 PM EDT)
Nyles Heise (10.03.2008 @04:53 AM EDT)
Hongcheng Zhu (10.03.2008 @07:58 AM EDT)
Joseph DeVincentris (10.03.2008 @10:28 AM EDT)
Wu Hao (10.03.2008 @10:28 AM EDT)
Jonathan Campbell (10.03.2008 @11:58 AM EDT)
Frank Yang (10.03.2008 @12:04 PM EDT)
Luke Pebody (10.03.2008 @12:17 PM EDT)
Gary M Gerken (10.03.2008 @01:11 PM EDT)
Krunoslav Kovac (10.03.2008 @02:10 PM EDT)
Wolf Mosle (10.03.2008 @05:27 PM EDT)
Jesse Kolman (10.03.2008 @11:29 PM EDT)
Alvaro Begue and John Tromp (10.04.2008 @02:54 PM EDT)
Fred Batty (10.04.2008 @04:32 PM EDT)
Frank Vernaillen (10.04.2008 @05:51 PM EDT)
Wolfgang Kais (10.04.2008 @08:58 PM EDT)
Andreas Eisele (10.05.2008 @06:13 PM EDT)
John G. Fletcher (10.05.2008 @06:44 PM EDT)
Ariel Flat (10.06.2008 @03:06 AM EDT)
Michael Liepelt (10.06.2008 @01:38 PM EDT)
Albert Stadler (10.06.2008 @03:24 AM EDT)
Abhishek Kumar (10.06.2008 @04:21 PM EDT)
James Dow Allen (10.06.2008 @04:40 PM EDT)
Michel Speiser (10.06.2008 @04:57 PM EDT)
Ya Liu (10.06.2008 @04:59 PM EDT)
Arthur Breitman (10.06.2008 @05:11 PM EDT)
Muharrem Gorkem (10.06.2008 @06:47 PM EDT)
Yu Yat Hong (10.06.2008 @11:20 PM EDT)
Paul Shearer (10.06.2008 @12:47 PM EDT)
Dani Gluck (10.07.2008 @08:59 AM EDT)
C A Subramanian (10.07.2008 @12:57 PM EDT)
Mark Perkins (10.08.2008 @04:07 PM EDT)
Tomas Schiffauer (10.07.2008 @06:38 PM EDT)
Dmitry Litvinenko (10.09.2008 @05:49 AM EDT)
Javier Rodriguez Soler (10.08.2008 @09:21 PM EDT)
Zhou Guang (10.09.2008 @08:08 AM EDT)
Ben Tarlow (10.09.2008 @01:13 PM EDT)
Karl D'SOUZA (10.09.2008 @03:30 PM EDT)
David Meiri (10.09.2008 @10:54 PM EDT)
Maris van Sprang (10.10.2008 @08:45 AM EDT)
Uri Yanover (10.10.2008 @02:46 PM EDT)
Boris Nikolaus (10.13.2008 @08:50 PM EDT)
Alex Wagner (10.14.2008 @01:31 PM EDT)
Frank E. Mullin (10.14.2008 @11:59 PM EDT)
Mithil Ramteke (10.15.2008 @01:38 PM EDT)
Will Hasenplaugh (10.15.2008 @07:19 PM EDT)
Nancy Schwarzkopf (10.16.2008 @06:05 PM EDT)
Michael Schaaf (10.17.2008 @04:12 PM EDT)
Greg Janoe (10.17.2008 @06:56 PM EDT)
Rickard Bjorklund and Carl Hammarlund (10.18.2008 @3:17 AM EDT)
David Friedman (10.18.2008 @09:43 PM EDT)
Xiaoyang Guan (10.20.2008 @07:14 PM EDT)
Martin Giese (10.21.2008 @11:16 AM EDT)
Michael Traut (10.21.2008 @10:50 PM EDT)
Andea Andenna (10.22.2008 @08:54 PM EDT)
Amos Guler (10.23.2008 @10:51 AM EDT)
Martin Thiim (10.24.2008 @05:59 AM EDT)
Naji Ashry (10.24.2008 @10:38 AM EDT)
Jimmy He (10.25.2008 @03:05 AM EDT)
Ashish Mishra (10.27.2008 @08:28 PM EDT)
Jan van Delden (10.28.2008 @09:03 AM EDT)
Jiri Navratil (10.28.2008 @08:40 PM EDT)
Radu Borza (10.29.2008 @11:22 AM EDT)
Jean-Pascal Harroch and Travis Craig (10.30.2008 @06:18 PM EDT)
Kai Guttmann (10.30.2008 @7:10 PM EDT)
Joyo Caldeira (10.30.2008 @07:36 PM EDT)
Timothy Eller (10.31.2008 @06:41 AM EDT)
Qin Chen (10.31.2008 @07:56 PM EDT)
Mate Bartalos (10.31.2008 @08:28 PM EDT)

Attention: If your name is posted here and you wish it removed please send email to the ponder@il.ibm.com.