Ponder This Challenge - July 2017 - Zeros in sin(PONDER)*sin(THIS)

Ponder This Challenge:

This month's challenge is based on an idea from Omid Motahed (thanks!).

Find a way to assign a unique digit to each letter such that the 863,172nd digit after the decimal point of sin(PONDER) and of sin(THIS) are not zeros, but more than a fifth of the first million digits of sin(PONDER)*sin(THIS) are zeros.

The parameters are in degrees, and we count the digits from 1, so for example, sin(100) = 0.9848077530122... so the fifth digit after the decimal point of sin(100) is zero.

Solution

• Click here to view the solution

  This solution is based on Oleg Vlasii's solution to the challenge.

  First, note that almost all numbers of the form sin(a)*sin(b) are irrational, and there is a very small probability of seeing the digit '0' more than 20%. Actually, Andreas Stiller checked and found that the irrational product having the most number of zeros in the first one million digits is

  sin(73°)*sin(75°) = 0.92371946158776085019620980936715820661197418131312... has only 101,008 zeros.

  But there are a few such products that are rational numbers:

  sin(15)*sin(75),
  sin(18)*sin(54),
  sin(30)*sin(30),
  sin(45)*sin(45),
  sin(60)*sin(60).
  

  To check the 863,172th digit, you can use the GMP library or simply ask WolframAlpha:
  "863173th digit of sin(18*pi/180)" ?

  sin(15), sin(30), sin(45), sin(60), and sin(75) have an 863172th digit of "0"; only sin(18) and sin(54) have "9" as their 863172th digit.
  So, one can choose PONDER = 18+360*k (or 180-18+360*k); THIS = 54+360*m (or 180-54+360*k); for example, PONDER = 902538, THIS = 7614.

  Thanks to Motty Porat for pointing us to the relevant reference: When is the (co)sine of a rational angle equal to a rational number? by Jorg Jahnel

Solvers

• Oleg Vlasii (28/06/17 03:21 PM IDT)
• Todd Will (28/06/17 04:05 PM IDT)
• Robert Gerbicz (28/06/17 04:13 PM IDT)
• Arthur Vause (28/06/17 09:33 PM IDT)
• Uoti Urpala (28/06/17 10:02 PM IDT)
• Tim Lewis (28/06/17 11:53 PM IDT)
• Bert Dobbelaere (29/06/17 12:38 AM IDT)
• Jesse Rearick (29/06/17 02:23 AM IDT)
• YanWu-He (29/06/17 06:56 AM IDT)
• Steven Langerwerf (29/06/17 08:38 IDT)
• Alper Halbutogullari (29/06/17 09:08 AM IDT)
• Rickard Björklund (29/06/17 10:10 AM IDT)
• Dan Dima (29/06/17 04:41 PM IDT)
• Alexandr Kozlinskiy (29/06/17 07:59 PM IDT)
• Alex Fleischer (29/06/17 08:18 PM IDT)
• Stéphane Higueret (29/06/17 08:19 PM IDT)
• Sergey Redyuk & Erik Lagerstedt (30/06/17 10:50 AM IDT)
• George Barnett (30/06/17 03:29 PM IDT)
• Daniëlle van den Berg (30/06/17 05:43 PM IDT)
• Teake Nutma (30/06/17 09:21 PM IDT)
• Luca Pintori (30/06/17 09:41 PM IDT)
• Rogerio Ponce da Silva (01/07/17 03:00 AM IDT)
• Tyler Mullen (01/07/17 03:09 AM IDT)
• Serge Batalov (01/07/17 06:36 AM IDT)
• Dmitry Kim (01/07/17 01:48 PM IDT)
• Andreas Stiller (01/07/17 05:06 PM IDT)
• Joseph DeVincentis (01/07/17 07:13 PM IDT)
• Hugo Pfoertner (01/07/17 10:42 PM IDT)
• David Grabner & Nikolay Gromov (01/07/17 11:20 PM IDT)
• Reiner Martin (02/07/17 01:27 AM IDT)
• Wolfgang Kais (02/07/17 03:21 AM IDT)
• Gary M. Gerken (02/07/17 08:21 AM IDT)
• Vladimir M. Milovanović (02/07/17 03:15 PM IDT)
• Aviv Nisgav (03/07/17 03:49 PM IDT)
• Amos Guler (03/07/17 10:50 PM IDT)
• Shirish Chinchalkar (03/07/17 10:51 PM IDT)
• Kurt Foster (04/07/17 04:10 PM IDT)
• Armin Krauß (04/07/17 07:41 PM IDT)
• David Greer (05/07/17 01:56 AM IDT)
• Harald Bögeholz (06/07/17 04:15 AM IDT)
• Chuck Carroll (06/07/17 01:32 PM IDT)
• Andreas Knüpfer (06/07/17 07:09 PM IDT)
• Arthur Rubin (07/07/17 02:27 AM IDT)
• Dawsen Hwang (08/07/17 05:43 AM IDT)
• Serge Thill (10/07/17 02:08 PM IDT)
• Stephen Lesko (12/07/17 12:44 AM IDT)
• Dieter Beckerle (12/07/17 02:33 PM IDT)
• Muralidhar Seshadri (12/07/17 07:04 PM IDT)
• Motty Porat (15/07/17 11:31 PM IDT)
• Erik Wünstel (16/07/17 06:04 PM IDT)
• Jean Buff (20/07/17 07:41 PM IDT)
• Vladimir Volevich (21/07/17 03:44 PM IDT)
• Michael J. Swart (21/07/17 11:42 PM IDT)
• Shouky Dan & Tamir Ganor (24/07/17 11:35 AM IDT)
• Tilo Anschütz (25/07/17 02:54 PM IDT)
• Peter Gerritson (27/07/17 04:20 AM IDT)
• Leonid Kalmankin (27/07/17 11:03 PM IDT)
• Greg Janée (28/07/17 08:37 AM IDT)
• Liubing Yu (29/07/17 11:34 PM IDT)
• Eden Saig (30/07/17 12:50 AM IDT)
• Kang Jin Cho (31/07/17 10:01 PM IDT)
• Oscar Volpatti (01/08/17 05:37 AM IDT)
• Peter Huggins (02/08/17 07:06 PM IDT)