Ponder This

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November 2022 - Challenge

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A sock drawer has b socks of exactly two types: a comfortable socks, and b-a non-comfortable socks. When choosing socks, one picks two socks at random (uniformly) from the drawer.

If a=15 and b=21, the probability of choosing two comfortable socks is exactly \frac{1}{2}. Assuming the sock drawer is magical and can contain a=568345 and b=803761, we also get the probability of \frac{1}{2}.

Your goal: find a,b such that the probability for two comfortable socks is exactly \frac{1}{974170}, and such that this is the minimal solution with b having at least 100 digits (minimal with respect to the size of b).

A bonus "*" will be given for finding the minimal D such that there are exactly 16 pairs a,b giving the probability \frac{1}{D} with b having less than 100 digits.

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: 27/10/2022 @ 16:30:00 PM EST
Solution: 04/12/2022 @ 12:00 PM EST
List Updated: 02/11/2022 @ 12:05 PM EST

People who answered correctly:

Paul Revenant (30/10/2022 2:40 PM IDT)
Lazar Ilic (30/10/2022 3:24 PM IDT)
Gary M. Gerken (30/10/2022 5:16 PM IDT)
*Alper Halbutogullari (31/10/2022 8:47 AM IDT)
*Giorgos Kalogeropoulos (31/10/2022 12:41 PM IDT)
Christoph Baumgarten (31/10/2022 6:52 PM IDT)
*Bertram Felgenhauer (31/10/2022 11:12 PM IDT)
Bert Dobbelaere (1/11/2022 12:29 AM IDT)
*Victor Chang (1/11/2022 6:38 AM IDT)
Richard Gosiorovsky(1/11/2022 12:22 PM IDT)
Harald Bögeholz (1/11/2022 3:49 PM IDT)
Lorenzo Gianferrari Pini (1/11/2022 6:35 PM IDT)
*Kurt Foster (2/11/2022 4:38 PM IDT)
*Stéphane Higueret (2/11/2022 4:39 PM IDT)
Josh Marza (2/11/20225:29 PM IDT)
Dan Dima (2/11/2022 7:10 PM IDT)
Davide Colì (2/11/2022 11:20 PM IDT)
*Prashant Wankhede (3/11/2022 5:31 AM IDT)
Michael Liepelt (3/11/2022 9:21 AM IDT)
Quentin Higueret (3/11/2022 2:25 PM IDT)
Lala Surya (4/11/2022 4:24 PM IDT)
Kang Jin Cho (4/11/2022 7:55 PM IDT)
*Franciraldo Cavalcante (4/11/2022 8:02 PM IDT)
*Sanandan Swaminathan (4/11/2022 10:16 PM IDT)
Aaron Kim (5/11/2022 4:14 PM IDT)
*Tim Walters (7/11/2022 1:12 AM IDT)
A. Ramanujan (7/11/2022 7:57 AM IDT)
Wilder Boyden (7/11/2022 5:55 PM IDT)
*David Greer (7/11/2022 6:05 PM IDT)
Mark Beyleveld (8/11/2022 4:17 PM IDT)
Laurence Warne (8/11/2022 6:33 PM IDT)
Latchezar Christov (8/11/2022 6:41 PM IDT)