Ponder This Challenge - August 2012 - 3 charges that almost cancel out

Ponder This Challenge:

Let C be the set of points with integer coordinates whose distance from the origin is no more than 1111 units.

Place three unit charges in three different quadrants in C, such that the equivalent force they apply on a unit charge at the origin is no more than 3E-12. For example, the three points (-2,-2), (-2,3) and (1,4) apply a force of ~0.121 on the charge at the origin.

Update 9/05: The best solution so far is ~4.008e-15, from Andreas Stiller; Jan Fricke; Vladimir Sedach; Peter Gerritson; Don Dodson; Mark Mammel; Tom Sirgedas; Jamie Jorgensen & Jason Lee; Patnaik Yasodhar, Siegbert Steinlechner; Arthur Breitman; Armin Krauss; Joël Bleuse; Giorgio Antonelli; Bradley Sherman; Arthur Vause; Deron Stewart; Anunay Kulshrestha; David Greer; Daniel Lecina; Shouky Dan & Tamir Ganor; Ioan Marius Popdan; Mark Pervovskiy; Andrea Andenna; Shilei Zang; Vincenzo Gianferrari Pini & Lorenzo Gianferrari Pini; Dan Dima and Adam Daire.

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

Solution

• Click here to view the solution

  One can get to a zero force by choosing three co-linear points, for example (12,12); (-15,-15); and (-20,-20). That is why we asked for the three points to be from three different quadrants.

  Another direction is to take an equilateral triangle and perturbate it a little to fit the integer lattice; for example (197,993); (-959,-326); and (761,-668), which gives a force of 1.6e-12.

  Another way to solve it is by using the template (x,-y); (-y,x); and (z,z), which gives the best solution (1074,-255); (-255,1074); and (762,762), which gives a force of 6e-15.

  Going over all the possible points in the sphere, we get (411,987); (-321,688); and (106,-635), for a force of 4e-15.

  We were looking for points inside the quadrants, i.e., not on the axes, but five of you (Kurt Hectic; Jo�l Bleuse; Deane Stewart; Arthur Breitman; and Hugo Pfoertner) sent us the solution (0, -814); (-608, 197); (608, 197) which applies a force of 3.2e-15 and one can argue that you can choose either quadrant for the 0,-814 point.

  Multiplying a given solution by a constant reduces the force, which is why we added the sphere constraint. If you ignore it - can we get to a zero force?

  ---

  If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Solvers

• Peter de Rivaz (08/01/2012 05:34 PM EDT)
• Chuck Carroll (08/01/2012 08:32 PM EDT)
• Deane Stewart (08/01/2012 10:07 PM EDT)
• Dario Serino (08/02/2012 01:58 AM EDT)
• Vladimir Sedach (08/02/2012 10:03 AM EDT)
• Hannes Schenck (08/02/2012 11:20 AM EDT)
• Chris Lewis (08/02/2012 12:21 PM EDT)
• Deron Stewart (08/02/2012 01:29 PM EDT)
• Amos Guler (08/02/2012 01:44 PM EDT)
• Arthur Breitman (08/02/2012 04:18 PM EDT)
• Adrian Orzepowski (08/02/2012 04:19 PM EDT)
• Christian Blatter (08/02/2012 05:39 PM EDT)
• Albert Stadler (08/02/2012 06:01 PM EDT)
• Kevin Sham (08/02/2012 06:06 PM EDT)
• Andrea Andenna (08/02/2012 06:19 PM EDT)
• Joseph DeVincentis (08/02/2012 09:52 PM EDT)
• Peter Park (08/02/2012 08:00 PM EDT)
• Dennis Langdeau (08/03/2012 12:17 AM EDT)
• Kenneth Meehan (08/03/2012 01:30 AM EDT)
• Jens Voss (08/03/2012 09:01 AM EDT)
• Stéphane Higueret (08/03/2012 10:26 AM EDT)
• Andreas Stiller (08/03/2012 01:52 PM EDT)
• Frederik Kaster (08/03/2012 02:02 PM EDT)
• Joshua Small (08/03/2012 04:34 PM EDT)
• William Wu & Matthew Klimesh (08/03/2012 06:06 PM EDT)
• Øyvind Grotmol (08/03/2012 06:07 PM EDT)
• Harald van Dijk (08/03/2012 07:05 PM EDT)
• Piotr Praczyk (08/03/2012 08:05 PM EDT)
• Don Dodson (08/03/2012 08:26 AM EDT)
• John Snyder (08/04/2012 12:25 AM EDT)
• Daniel Branscombe (08/04/2012 09:45 AM EDT)
• Andy Carle (08/04/2012 02:37 PM EDT)
• Todd Will (08/04/2012 04:05 PM EDT)
• Arthur Vause (08/05/2012 02:27 AM EDT)
• Reinder Verlinde (08/05/2012 08:19 AM EDT)
• Bruce Fleischer (08/05/2012 11:20 AM EDT)
• Christian Pape (08/05/2012 12:57 PM EDT)
• Maris Van Sprang (08/05/2012 12:58 PM EDT)
• Harald Bögeholz (08/05/2012 04:45 PM EDT)
• Mark Pervovskiy (08/05/2012 10:42 PM EDT)
• Liubing Yu (08/06/2012 04:07 AM EDT)
• Markus von Rimscha (08/06/2012 01:15 PM EDT)
• James Dow Allen (08/06/2012 01:16 AM EDT)
• Fletcher Dostie (08/06/2012 01:42 PM EDT)
• Mark Mammel (08/06/2012 01:50 PM EDT)
• Armin Krauss (08/06/2012 05:45 PM EDT)
• Sergey Grishaev (08/06/2012 06:17 PM EDT)
• Adir HaCohen (08/07/2012 06:25 AM EDT)
• Jan Fricke (08/07/2012 06:58 AM EDT)
• Shilei Zang (08/07/2012 09:30 AM EDT)
• Walter Schmidt (08/07/2012 12:54 PM EDT)
• Daniel Bitin (08/07/2012 01:03 PM EDT)
• Vitor Miguel (08/07/2012 05:46 PM EDT)
• Wuhao (08/08/2012 12:17 PM EDT)
• Shelby Doolittle (08/08/2012 02:42 PM EDT)
• Yasodhar (Pattu) Patnaik (08/08/2012 06:35 PM EDT)
• Peter Gerritson (08/09/2012 12:51 AM EDT)
• Vincent Lejeune (08/09/2012 09:55 AM EDT)
• Adam Daire (08/09/2012 12:11 PM EDT)
• Duane Bailey (08/09/2012 02:31 PM EDT)
• David Meiri (08/09/2012 02:54 PM EDT)
• Michael Lim (08/10/2012 12:22 AM EDT)
• Tom Sirgedas (08/10/2012 01:39 AM EDT)
• Fabio Negroni (08/10/2012 02:55 AM EDT)
• Ariel Flat (08/10/2012 11:37 AM EDT)
• Allen Welkie (08/10/2012 05:48 PM EDT)
• Jamie Jorgensen & Jason Lee (08/10/2012 08:41 PM EDT)
• Hakan Summakoglu (08/12/2012 09:09 AM EDT)
• Joël Bleuse (08/12/2012 06:46 PM EDT)
• Patnaik Yasodhar (08/13/2012 12:00 AM EDT)
• Don Ryan (08/13/2012 12:57 AM EDT)
• Siegbert Steinlechner (08/13/2012 03:32 AM EDT)
• Rob Pratt (08/13/2012 11:49 AM EDT)
• Joaquim Neves Carrapa (08/09/2012 07:20 AM EDT)
• David Blackston (08/13/2012 07:50 PM EDT)
• Guangda Huzhang (08/15/2012 12:46 AM EDT)
• Thomas Egense (08/15/2012 03:12 AM EDT)
• Heiko Selber (08/15/2012 12:47 PM EDT)
• Daniel Lo (08/15/2012 08:10 PM EDT)
• Giorgio Antonelli (08/16/2012 03:20 AM EDT)
• Ryo Sugihara (08/16/2012 04:31 AM EDT)
• Javier Rodriguez (08/16/2012 06:26 AM EDT)
• Hyunkee Sung (08/16/2012 02:56 PM EDT)
• Bradley Sherman (08/16/2012 05:33 PM EDT)
• Anoop Ghanwani (08/16/2012 08:06 PM EDT)
• Fred Schalekamp (08/18/2012 06:13 PM EDT)
• Seung Min "Brian" Park (08/19/2012 10:45 PM EDT)
• Anunay Kulshrestha (08/21/2012 06:34 AM EDT)
• Dan Dima (08/21/2012 09:07 AM EDT)
• Chris Becker (08/21/2012 09:35 AM EDT)
• Francis Golding (08/21/2012 01:24 PM EDT)
• Matej Kollár (08/22/2012 10:35 AM EDT)
• David Greer (08/23/2012 10:32 AM EDT)
• Daniel Lecina (08/23/2012 12:51 PM EDT)
• Mutaamba Maasha (08/24/2012 10:40 PM EDT)
• Shouky Dan & Tamir Ganor (08/25/2012 12:50 PM EDT)
• Rami Pearl (08/25/2012 06:51 PM EDT)
• Stanislav Grudnitskiy (08/26/2012 04:14 AM EDT)
• Phil Benamy & John Bortins (08/26/2012 12:50 PM EDT)
• Ioan Marius Popdan (08/27/2012 03:34 AM EDT)
• Tony Harrison (08/27/2012 05:57 PM EDT)
• Ming Song (08/29/2012 04:46 PM EDT)
• Radu-Alexandru Todor (08/31/2012 06:27 AM EDT)
• Vincenzo Gianferrari Pini & Lorenzo Gianferrari Pini (08/31/2012 06:27 AM EDT)
• Rainer Rosenthal (08/31/2012 01:25 PM EDT)
• Mauro Bampo (08/31/2012 06:16 PM EDT)
• Alex Wagner (09/01/2012 12:05 AM EDT)
• Dan Ismailescu (09/05/2012 09:23 PM EDT)