Ponder This Challenge - October 2011 - Interesting resistors network

Ponder This Challenge:

We call a real number R "interesting" if the product of every 8 consecutive decimal digits is 40,320.

Build a network from no more than twenty 1-ohm resistors such that their overall resistance, see (external link http://en.wikipedia.org/wiki/Series_and_parallel_circuits for details) R would be interesting.

Please submit your answer as pairs of letters, in which each pair represents the endpoints of a resistor, where the overall resistance is measured between "a" and "z".

For example, the following network is a cube, measured along the long diagonal: "ab bc cd da ef fz zh he ae bf cz dh".

We know that atto-ohm accuracy is meaningless, but this is a mathematical challenge, not an electrical engineering one.

Hint: This month's challenge can be solved on the back of an envelope.

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!

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Solution

• Click here to view the solution

  An interesting number, as we defined it, must have a cycle of 8 in its decimal representation.

  A cycle 8 number can be represented as a rational number with denominator 99999999.

  If the cycle is 2 or 4, the number cannot be interesting, since 40320 is not a square of an integer.

  Since we are looking for numbers that are not cycle-2 or cycle-4, it must have the form of X/10001.

  10001=137*73, and therefore we can check X/73 and X/137.

  The first always has a zero in the 8-digits cycle, which rules it out.

  The second has several interesting cases.

  Using a continued fraction, we can represent them as a network of less than 20 resistors.

  The smallest parallel and series combination solution uses only 12 resistors: ab bz bz bz ac cz cz cd dz ce ef fz, which gives an overall resistance of 92/137; but if you allow for general network than (thanks to Armin Krauss) you can have 11 resistors only: "ab bc cz ad dz bd az az ae ez ez".

  If you want to insist that *every* 8 consecutive digits has a product of 40,320, then there exists a 15-resistor solution (ab bc cd de ef fg gz gh hi iz hz hz hz aj jz), which has an overall resistance of 210/137; or even better: (thanks to Mathias Schenker): only 13 resistors "ab bc cd de ef fg gz gh hi iz hz hz hz aj jz".

  Piet Orye noted that 53 resistors suffice to solve a variant where we define interesting as having all 10 consecutive digits contain all 10 decimal digits.

Solvers

• Jan Fricke (09/30/2011 08:32 PM EDT)
• János Kramár (10/01/2011 12:22 AM EDT)
• Yuanmi Chen (10/01/2011 08:27 AM EDT)
• Alan Murray (10/01/2011 11:44 AM EDT)
• Dan Dima (10/01/2011 12:54 PM EDT)
• Leif Jensen (10/01/2011 01:22 PM EDT)
• Pei Wu (10/02/2011 11:17 AM EDT)
• Lu Wang (10/02/2011 12:04 AM EDT)
• Piet Orye (10/02/2011 06:00 AM EDT)
• Peter Gerritson (10/02/2011 07:26 PM EDT)
• Ariel Flat (10/02/2011 08:19 PM EDT)
• Arthur Breitman (10/03/2011 12:36 PM EDT)
• Harold Gutch & Stefan Zehl (10/03/2011 01:15 PM EDT)
• William Mantzel (10/03/2011 02:00 PM EDT)
• Andreas Razen (10/03/2011 03:08 PM EDT)
• Radu-Alexandru Todor (10/03/2011 07:21 PM EDT)
• Chuck Carroll (10/03/2011 08:05 PM EDT)
• Avishalom Shalit (10/04/2011 11:33 AM EDT)
• Chris Kuklewicz (10/04/2011 04:58 PM EDT)
• Adam Daire (10/04/2011 07:04 PM EDT)
• Victor Chang (10/04/2011 08:49 PM EDT)
• Josh Small (10/06/2011 11:54 AM EDT)
• Ken Bateman (10/06/2011 02:40 PM EDT)
• Shilei Zang (10/06/2011 03:46 PM EDT)
• Yasodhar Patnaik (10/07/2011 03:26 AM EDT)
• Gary M. Gerken (10/08/2011 08:01 PM EDT)
• Renato Fonseca (10/09/2011 03:19 PM EDT)
• David Greer (10/10/2011 07:45 AM EDT)
• John Tromp (10/10/2011 12:50 PM EDT)
• Maohua Sheng (10/10/2011 12:52 PM EDT)
• Daniel Bitin (10/11/2011 11:07 AM EDT)
• Phil Muhm (10/11/2011 02:53 PM EDT)
• Armin Krauss (10/11/2011 03:14 PM EDT)
• Liubing Yu (10/11/2011 06:30 PM EDT)
• Adrian Neacsu (10/12/2011 04:21 PM EDT)
• Hamidreza Bidar (10/12/2011 06:15 PM EDT)
• Clive Tong (10/13/2011 05:09 PM EDT)
• Movin Jain (10/14/2011 01:26 PM EDT)
• Joseph DeVincentis (10/14/2011 10:08 PM EDT)
• Pritibhushan Sinha (10/15/2011 04:48 AM EDT)
• Peter de Rivaz (10/15/2011 04:59 AM EDT)
• Dejan Stakic (10/16/2011 06:46 AM EDT)
• David Friedman (10/16/2011 11:05 AM EDT)
• Manoj vedvistara (10/16/2011 01:23 PM EDT)
• Harald Bögeholz (10/16/2011 03:17 PM EDT)
• Jeff Steele (10/17/2011 03:06 PM EDT)
• Kipp Johnson & Nick Vigo (10/18/2011 11:58 AM EDT)
• Tom Sirgedas (10/18/2011 08:45 PM EDT)
• Chris Shannon (10/19/2011 01:04 AM EDT)
• Jan Braunisch (10/19/2011 02:51 AM EDT)
• Dr. Beena George (10/19/2011 05:02 AM EDT)
• Stéphane Higueret (10/20/2011 10:07 PM EDT)
• Franza Cavalcante Jr (10/21/2011 03:20 PM EDT)
• Haoyu Bai (10/25/2011 01:01 AM EDT)
• Thomas Rohr (10/26/2011 05:19 AM EDT)
• Mathias Schenker (10/27/2011 11:10 AM EDT)
• Yuan Yuan (10/28/2011 11:42 PM EDT)
• Frederik Kaster (10/31/2011 03:02 PM EDT)