Ponder This Challenge - May 2006 - Distance configuration of 4 points

Ponder This Challenge:

Puzzle for May 2006.

This month's puzzle concerns configurations of 4 distinct points in the plane.  Suppose we know the 6 distances between pairs of points only take on two values.  We ask for a list of the distinct possible configurations.  For each configuration give the number of long distances and the number of short distances (which must sum to 6) and the ratio of the long distance to the short distance.

The first 100 people who answer all parts correctly will be listed. The answer will be posted a week after the 100th is received, or at the end of the month.

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

  Answer:

  The possible configurations can be listed as follows:

  1.  5 short, 1 long, ratio sqrt(3)=1.73201...
  2.  4 short, 2 long, ratio sqrt(2)=1.41421...
  3.  4 short, 2 long, ratio sqrt(2+sqrt(3))=1.93185...
  4.  3 short, 3 long, ratio sqrt(3)=1.73201...
  5.  3 short, 3 long, ratio .5*(1+sqrt(5))=1.61803...
  6.  2 short, 4 long, ratio sqrt(2+sqrt(3))=1.93185...

  They can be described as follows:

  1.  Two equilateral triangles sharing an edge.
  2.  Square
  3.  Isosceles triangle with center point
  4.  Equilateral triangle with center point
  5.  Regular pentagon with a point deleted
  6.  Kite

  This problem was provided to me by John Fletcher.  It has appeared in several places before and is apparently originally due to Einhorn and Schoenberg.

  Since this was a fairly easy problem I only gave credit for solutions which were completely correct.  There were some additional solutions which were largely correct.

  ---

  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

• Dan Dima (05.02.2006 @10:30:04 AM EDT)
• Joseph DeVincentis (05.02.2006 @10:41:23 AM EDT)
• Balakrishnan V (05.02.2006 @11:46:58 AM EDT)
• Shuo Pan (05.02.2006 @02:19:10 PM EDT)
• Frank Mullin (05.02.2006 @02:37:41 PM EDT)
• John Hart (05.02.2006 @03:24:16 PM EDT)
• Joe BGI SF Fendel (05.02.2006 @04:15:11 PM EDT)
• Graeme McRae (05.02.2006 @06:04:22 PM EDT)
• Mark Perkins (05.02.2006 @06:39:19 PM EDT)
• Wu Hao (05.02.2006 @11:10:05 PM EDT)
• David W. Cantrell (05.03.2006 @12:25:51 AM EDT)
• aman_cc (05.03.2006 @12:35:58 PM EDT)
• Li Han (05.03.2006 @01:36:15 PM EDT)
• John Dalbec (05.03.2006 @02:59:29 PM EDT)
• Dinesh Krithivasan (05.03.2006 @04:31:47 PM EDT)
• Lukas Saul (05.04.2006 @02:58:26 AM EDT)
• Michael Brand (05.04.2006 @03:35:44 AM EDT)
• Biantaishab (05.04.2006 @07:32:28 AM EDT)
• Eugene Vasilchenko (05.04.2006 @10:02:58 AM EDT)
• Kaikai Shen (05.04.2006 @11:26:50 AM EDT)
• Harold Gutch (05.04.2006 @11:55:30 AM EDT)
• Matthew Charlap (05.04.2006 @01:30:44 PM EDT)
• David P Stigant (05.04.2006 @06:17:01 PM EDT)
• William C Hasenplaugh (05.04.2006 @06:21:34 PM EDT)
• Sandeep Rathour (05.05.2006 @12:58:17 AM EDT)
• Se Kwon Kim (05.05.2006 @02:29:04 AM EDT)
• Alex Izvalov (05.05.2006 @03:41:30 AM EDT)
• Roberto Tauraso (05.05.2006 @05:21:52 AM EDT)
• Miroslava Sotakova (05.05.2006 @05:36:24 AM EDT)
• Edna Mantsour (05.05.2006 @06:07:57 AM EDT)
• David McQuillan (05.05.2006 @07:02:58 AM EDT)
• Firat Solgun (05.05.2006 @07:10:57 AM EDT)
• Keith Anker (05.05.2006 @08:59:35 AM EDT)
• Greg Bubnis (05.05.2006 @09:13:15 AM EDT)
• Phil Muhm (05.05.2006 @09:51:55 AM EDT)
• James Dow Allen (05.05.2006 @10:51:46 PM EDT)
• Pratik Dixit (05.06.2006 @06:32:14 AM EDT)
• Philippe Fondanaiche (05.06.2006 @06:51:05 AM EDT)
• Anonymous (05.07.2006 @03:20:35 AM EDT)
• Jesse Blosser (05.07.2006 @12:21:59 PM EDT)
• Dmytry Lavrov (05.07.2006 @01:17:35 PM EDT)
• Gary M Gerken (05.07.2006 @05:19:18 PM EDT)
• R. Nandakumar (05.08.2006 @02:06:22 AM EDT)
• Francis Golding (05.08.2006 @05:07:11 PM EDT)
• Deron Stewart (05.08.2006 @11:10:08 PM EDT)
• Anneke Van Steirteghem (05.09.2006 @06:09:49 AM EDT)
• Joël Bleuse (05.09.2006 @12:05:44 PM EDT)
• Andrew Buchanan (05.09.2006 @01:31:31 PM EDT)
• Todd M Pocklington (05.10.2006 @09:33:50 AM EDT)
• Yoav Raz (05.10.2006 @02:47:20 PM EDT)
• Wolfgang Kais (05.10.2006 @06:14:54 PM EDT)
• Chuck Carroll (05.10.2006 @09:13:21 PM EDT)
• Jiri Hrdina (05.11.2006 @10:30:31 AM EDT)
• Zhou Guang (05.11.2006 @09:05:22 PM EDT)
• Leroy Junker (05.11.2006 @10:57:25 PM EDT)
• Gopalkrishna Hegde (05.12.2006 @03:22:45 AM EDT)
• Natalie Aisbett (05.12.2006 @05:48:50 AM EDT)
• P.J.M. Veugen (05.12.2006 @08:05:03 AM EDT)
• Edward Blakey (05.12.2006 @09:24:13 AM EDT)
• Du Yang (05.12.2006 @08:43:26 PM EDT)
• Daniel Bitin (05.13.2006 @03:02:26 PM EDT)
• Claudio Baiocchi (05.15.2006 @03:33:52 AM EDT)
• Mike Bell (05.15.2006 @02:40:22 PM EDT)
• Dale Greenlee (05.17.2006 @09:22:59 PM EDT)
• Shanmukh Katragadda (05.19.2006 @06:28:28 AM EDT)
• Daniel Chong Jyh Tar (05.19.2006 @01:19:15 PM EDT)
• Kerry M Soileau (05.22.2006 @10:31:58 AM EDT)
• Jam (05.23.2006 @10:18:42 AM EDT)
• es Holland (05.23.2006 @10:18:42 AM EDT)
• Greg Janée (05.23.2006 @05:38:13 PM EDT)
• Kennan Shelton (05.23.2006 @06:13:12 PM EDT)
• Sridivakar Inakonda (05.24.2006 @10:33:16 AM EDT)
• Markus Nauroth (05.25.2006 @07:55:10 AM EDT)
• Vinoad Senguttuvan (05.25.2006 @10:04:10 AM EDT)
• Patrick Fleury (05.25.2006 @08:12:05 PM EDT)
• Chris Lomont (05.26.2006 @10:06:50 AM EDT)
• Federico (05.29.2006 @02:40:27 AM EDT)
• Jan Rubak (05.30.2006 @03:36:39 PM EDT)
• Dion K. Harmon (05.31.2006 @00:02:40 AM EDT)
• Jens Voß (05.31.2006 @04:47:02 AM EDT)
• Alan O'Donnell (05.31.2006 @09:54:52 AM EDT)
• Mike Bodell (06.01.2006 @12:29:01 AM EDT)
• Ed Rak (06.01.2006 @02:20:36 AM EDT)