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# Ponder This

Welcome to our monthly puzzles.
You are cordially invited to match wits with some of the best minds in IBM Research.

## November 2022 - Challenge

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: 19/12/2022 @ 15:05 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 (2/11/2022 5:29 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)
Keun Hyong Kwak (9/11/2022 3:03 AM IDT)
Dieter Beckerle (10/11/2022 12:08 PM IDT)
Daniel Bitin (10/11/2022 11:06 PM IDT)
Ian Lee (11/11/2022 2:28 AM IDT)
Lorenz Reichel (11/11/2022 10:08 AM IDT)
Teawoo Kim kim (12/11/2022 1:59 AM IDT)
Austin Lee (12/11/2022 6:20 PM IDT)
*Marco Bellocchi (13/11/2022 2:05 AM IDT)
Shouky Dan & Tamir Ganor (14/11/2022 3:04 PM IDT)
Karl Mahlburg (15/11/2022 4:52 AM IDT)
*Vladimir Volevich (15/11/2022 6:37 AM IDT)
Jesús Vergara Temprado (16/11/2022 3:39 PM IDT)
Daniel Chong Jyh Tar (17/11/2022 7:58 AM IDT)
Arthur Vause (17/11/2022 3:58 PM IDT)
*Richard T Liu (19/11/2022 11:03 PM IDT)
Kipp Johnson (20/11/2022 6:56 PM IDT)
Diana Gornea (22/11/2022 5:43 PM IDT)
Sachal Mahajan (23/11/2022 12:04 AM IDT)
Albert Stadler (24/11/2022 9:24 AM IDT)
*Nyles Heise (26/11/2022 8:16 AM IDT)
*Chris Shannon (27/11/2022 1:40 AM IDT)
*Alexander Gramolin (28/11/2022 6:11 AM IDT)
*Li Li (28/11/2022 7:54 AM IDT)
Carl Löndahl (28/11/2022 8:57 PM IDT)
Aaditya Raghavan (29/11/2022 1:56 AM IDT)
Ali Gurel (29/11/2022 8:52 AM IDT)
*Andreas Puccio (29/11/2022 9:25 PM IDT)
Radu-Alexandru Todor (30/11/2022 12:45 AM IDT)
David F.H. Dunkley (30/11/2022 1:28 AM IDT)
Reiner Martin (30/11/2022 6:04 PM IDT)
Oscar Volpatti (1/12/2022 3:00 AM IDT)
Motty Porat (2/12/2022 4:51 PM IDT)
Martin Bisson (4/12/2022 9:11 AM IDT)