Ponder This Challenge - November 2022 - The sock drawer

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.

Solution

• Click here to view the solution

  November 2022 Solution:

  One possible solution is

  a =
  578531456294544759193960639508933458692135986207286734779245556353330267012496957470063389129282
  41502
  b =
  532012360194801766547790141073187864842946628525748481714720078933587785103495458094301593151732
  52600116
  

  For the bonus question we have D = 614. One possible method of obtaining this solutions is by converting the equation $\frac{a(a-1)}{b(b-1)}=\frac{1}{N}$ to an equation of the form $x^2-dy^2=1$.
  Such an equation is called **Pell's equation** and there are standard method for solving it and generating all the solutions.

Solvers

• 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)