Ponder This Challenge - June 2013 - Find a fraction

Ponder This Challenge:

Find a rational number (a fraction of two integers) that satisfies the following conditions:

1. Its denominator is five digits long and all of them are different.
2. In the infinite decimal representation, every digit occurs an equal number of times in the digits from one billion to two billion places to the right of the decimal point (inclusive), except for the last digit in the denominator, which occurs twice as often as the other nine digits.
3. Its numerator contains as few different digits as possible.

Update 6/4: To clarify the problem, here is an example: 5/17 = 0.294117647058823529411764705882... and in the 11 digit in the places 20-30 after the decimal point the digit 7 (the last digit of the demonimator) appears twice as many times as the digit 4, but not twice as many as the digit 9 (which does not appear there at all).

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Solution

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  The way to have one digit appear twice as many times as the other nine is to set a period of 11 digits. To do that, we need the fraction to be X/99999999999. Factoring the denominator leads us to 21649 as the denominator, and looking at the possible numerators we find that 639/21649 solves the first two conditions.

  To get fewer number of distinct digits, 646/21649 may seem to be the best solution (there are only two different digits), but using improper fractions, we can get the best solution: 333333/21649.

Solvers

• Radu-Alexandru Todor (05/31/2013 07:00 PM EDT)
• Joseph DeVincentis (06/03/2013 03:33 PM EDT)
• John Tromp (06/03/2013 03:57 PM EDT)
• Lorenzo Gianferrari Pini (06/04/2013 10:02 AM EDT)
• Alex Wagner (06/04/2013 10:30 PM EDT)
• Arthur Miklos (06/05/2013 02:41 AM EDT)
• Phil Strenski (06/05/2013 12:47 PM EDT)
• Dan Dima (06/06/2013 04:49 PM EDT)
• José Eduardo Gaboardi de Carvalho (06/06/2013 08:19 PM EDT)
• Kan Shen (06/07/2013 02:02 PM EDT)
• Florian Fischer (06/07/2013 05:44 PM EDT)
• Dave Dodson & Don Dodson (06/07/2013 10:06 PM EDT)
• Vladimir Sedach (06/08/2013 02:44 AM EDT)
• Peter Holdsworth (06/08/2013 05:31 AM EDT)
• Daniel Bitin (06/08/2013 06:36 AM EDT)
• Brendan Berger (06/08/2013 09:26 PM EDT)
• Adrian Orzepowski (06/09/2013 04:11 PM EDT)
• David Greer (06/10/2013 01:29 PM EDT)
• Krunoslav Kovac (06/10/2013 02:56 PM EDT)
• Alex Fleischer (06/12/2013 05:17 AM EDT)
• Hugo Pfoertner (06/12/2013 03:10 PM EDT)
• Tim Cieplowski (06/11/2013 06:40 PM EDT)
• Kenneth Arnold & Jack Kennedy & Patrick Gibbons (06/11/2013 10:30 PM EDT)
• Antoine Comeau (06/12/2013 03:51 PM EDT)
• Dieter Beckerle (06/12/2013 04:17 PM EDT)
• Tom Harley (06/12/2013 04:45 PM EDT)
• Arthur Vause (06/12/2013 09:05 PM EDT)
• Javier Rodriguez (06/13/2013 05:48 AM EDT)
• Andreas Stiller (06/13/2013 06:47 AM EDT)
• Jan Fricke (06/13/2013 12:54 PM EDT)
• Motty Porat (06/13/2013 01:26 PM EDT)
• Chuck Carroll (06/14/2013 09:54 AM EDT)
• Shirish Chinchalkar (06/15/2013 06:03 AM EDT)
• Tony Harrison (06/15/2013 06:15 PM EDT)
• Dezhi Zhao (06/16/2013 02:36 AM EDT)
• Marco Bellocchi (06/16/2013 01:23 PM EDT)
• Franza Cavalcante Jr (06/18/2013 03:47 PM EDT)
• Pataki Gergely (06/18/2013 05:28 PM EDT)
• Dharmadeep Muppalla (06/19/2013 11:27 AM EDT)
• Hui Han Chin (06/19/2013 11:55 AM EDT)
• Levon Haykazyan (06/19/2013 07:24 PM EDT)
• Todd Will (06/21/2013 09:08 AM EDT)
• Harald Bögeholz (06/17/2013 04:35 PM EDT)
• Hakan Summakoglu (06/18/2013 06:22 AM EDT)
• Clive Tong (06/18/2013 12:14 PM EDT)
• Olivier Mercier (06/22/2013 02:27 AM EDT)
• Armin Krauss (06/22/2013 09:55 AM EDT)
• Amos Guler (06/23/2013 11:31 AM EDT)
• Gyorgy Gyomai (06/23/2013 03:45 PM EDT)
• Jeong Soo Kim (06/23/2013 04:18 PM EDT)
• Philippe Fondanaiche (06/24/2013 02:23 PM EDT)
• Liubing Yu (06/26/2013 03:44 AM EDT)
• Wanxin Yuan (06/27/2013 12:04 AM EDT)
• Seongwoo Hong (06/27/2013 09:13 AM EDT)
• Stéphane Higueret (06/28/2013 07:12 PM EDT)
• Michael Lim (06/28/2013 08:15 PM EDT)
• Albert Stadler (06/29/2013 01:20 AM EDT)
• Nivrutti Tambe (06/29/2013 09:43 PM EDT)
• Mauro Bampo (06/30/2013 05:39 AM EDT)
• Mark Pervovskiy (07/01/2013 07:59 AM EDT)