Ponder This Challenge - July 2007 - Minimum length of blocking fence

Ponder This Challenge:

Given a square piece of property of unit side you wish to build fences so that it is impossible to see through the property, ie there is no sightline connecting two points outside the property and passing through the property that does not intersect a fence. The fences do not have to be connected and several fences can come together at a point. What is the minimum total length of fencing required and how is it arranged. For example you could place fencing along all four sides. This would have total length 4 but is not the best possible.

I do not have a proof that my answer is minimal. I will accept as correct any answer that does at least as well.

UPDATE, 7/12/07: The fences can be placed in the interior of the property, they aren't restricted to the boundary.

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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

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  Ponder This Solution:

  Suppose the corners of the property are at (0,0), (1,0), (1,1) and (0,1). The best solution known to me is the following. Connect the point (.5-sqrt(3)/6,.5-sqrt(3)/6) by straight fences to the corners (0,0), (1,0) and (0,1). Connect the point (.5,.5) to the corner (1,1) with a straight fence. The total length is 2*sqrt(2/3)+(sqrt(2)/2 - sqrt(6)/6)+sqrt(2)/2 = (2/3)*sqrt(6)+(sqrt(2)/2)-sqrt(6)/6+sqrt(2)/2 = sqrt(2)+sqrt(6)/2 = 2.638958433764684+.

  A version of this problem appeared in the January 1983 issue of the "Tentacle" an internal Lawrence Livermore National Laboratory publication (p. 30 "Puzzles by Henry Larson" edited by Harry Nelson). I learned of it from John Fletcher.

  Thanks to some of the solvers for pointing me to additional appearances of this problem. The earliest I found was:
  R. E. D. Jones, "Opaque Sets of Degree alpha", American Mathematical Monthly, May 1964, p. 535-537.

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  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 (07.06.2007 @03:58:28 PM EDT)
• V Balakrishnan (07.06.2007 @07:32:09 PM EDT)
• Jiri Navratil (07.06.2007 @07:49:05 PM EDT)
• Celia Anteneodo (07.07.2007 @07:45:42 AM EDT)
• Wei Zhou (07.07.2007 @11:31:49 PM EDT)
• Andrew Buchanan (07.08.2007 @09:52:10 AM EDT)
• Henry Bottomley (07.09.2007 @03:57:54 AM EDT)
• Erik Hostens (07.09.2007 @11:08:31 AM EDT)
• Almudena Moratinos (07.09.2007 @11:54:23 AM EDT)
• Dion Harmon (07.09.2007 @12:24:56 PM EDT)
• Joe Riel (07.09.2007 @05:26:12 PM EDT)
• Michael Brand (07.09.2007 @08:20:03 PM EDT)
• Andreas Eisele (07.09.2007 @10:15:33 PM EDT)
• Eugene Vasilchenko (07.10.2007 @12:00:15 AM EDT)
• Vineet Gupta (07.10.2007 @06:30:53 AM EDT)
• Kerry M. Soileau (07.10.2007 @01:58:08 PM EDT)
• Phil Muhm (07.10.2007 @02:07:37 PM EDT)
• Josh Boorstein (07.10.2007 @03:05:57 PM EDT)
• William Hasenplaugh (07.10.2007 @05:12:58 PM EDT)
• Chris Shannon (07.10.2007 @05:18:44 PM EDT)
• Frank Yang (07.10.2007 @07:52:20 PM EDT)
• Ben Tarlow (07.11.2007 @03:46:55 AM EDT)
• Ashutosh Mehra (07.11.2007 @04:10:20 AM EDT)
• Nikhil Sahasrabudhe (07.11.2007 @04:11:55 AM EDT)
• Michael Swart (07.11.2007 @10:21:54 AM EDT)
• Michael Quist (07.11.2007 @10:27:22 AM EDT)
• Shanmukh Katragadda (07.11.2007 @12:35:05 PM EDT)
• Sanjeev Gupta (07.11.2007 @12:41:18 PM EDT)
• Daniel Hsu (07.11.2007 @11:10:12 PM EDT)
• Ed Shepard (07.12.2007 @12:34:01 AM EDT)
• Yu Yat Hong (07.12.2007 @02:02:39 AM EDT)
• James Dow Allen (07.12.2007 @03:11:06 AM EDT)
• Albert Stadler (07.12.2007 @04:51:07 AM EDT)
• Alfonso Nishikawa (07.12.2007 @07:21:43 AM EDT)
• Anton Luht (07.12.2007 @10:01:26 AM EDT)
• Mialja Machiama (07.12.2007 @10:55:08 AM EDT)
• Marissa Wilk (07.12.2007 @11:03:02 AM EDT)
• Graeme McRae (07.12.2007 @11:59:55 AM EDT)
• Arno Candel (07.12.2007 @01:45:28 PM EDT)
• Michael Slifker (07.12.2007 @03:03:18 PM EDT)
• Chris Lomont (07.12.2007 @03:28:58 PM EDT)
• Alan O'Donnell (07.12.2007 @04:10:40 PM EDT)
• William Magaletta (07.12.2007 @07:28:52 PM EDT)
• Lizhao Zhang (07.12.2007 @08:50:16 PM EDT)
• Randall Szage and Gary Gerken (07.13.2007 @03:19:54 AM EDT)
• Joel Bleuse (07.13.2007 @11:01:36 AM EDT)
• Christian Blatter (07.13.2007 @11:20:57 AM EDT)
• Gerald Pieris (07.13.2007 @02:14:25 PM EDT)
• Jesse G. Beu (07.13.2007 @02:46:18 PM EDT)
• Majid Khabbazian (07.13.2007 @04:12:48 PM EDT)
• Heiko Selber (07.13.2007 @05:12:04 PM EDT)
• Arthur Breitman (07.13.2007 @06:06:17 PM EDT)
• Dave Parashar (07.14.2007 @01:16:52 AM EDT)
• Gale Greenlee (07.15.2007 @04:37:00 PM EDT)
• David Greer (07.15.2007 @05:02:53 PM EDT)
• Adel El-Atawy (07.16.2007 @05:16:09 AM EDT)
• Wolf Mosle (07.16.2007 @05:03:01 PM EDT)
• Josh Jaffe (07.16.2007 @07:56:00 PM EDT)
• Se Kwon Kim (07.17.2007 @01:19:56 PM EDT)
• George Saon (07.17.2007 @03:03:51 PM EDT)
• Yurun Liu (07.17.2007 @03:46:27 PM EDT)
• Chris Archer (07.17.2007 @04:14:15 PM EDT)
• David McQuillan (07.17.2007 @04:17:39 PM EDT)
• Chris & Richard Hills (07.17.2007 @04:56:22 PM EDT)
• Maxim G. Smith (07.17.2007 @09:17:56 PM EDT)
• Georg Wille (07.18.2007 @03:14:18 AM EDT)
• Mauro Piazza (07.18.2007 @11:25:42 AM EDT)
• Brian Puchala (07.18.2007 @02:21:25 PM EDT)
• Chuck Grant (07.18.2007 @02:31:08 PM EDT)
• Jonathan Gu (07.18.2007 @03:04:18 PM EDT)
• Johan de Ruiter (07.18.2007 @04:13:03 PM EDT)
• Niraj Jha (07.18.2007 @05:16:36 PM EDT)
• Andrew Jones (07.19.2007 @09:08:07 AM EDT)
• Rogerio Ponce (07.19.2007 @11:03:35 AM EDT)
• Biju Mohan (07.19.2007 @12:13:53 PM EDT)
• Manish Katiyar (07.19.2007 @01:07:48 PM EDT)
• Brijesh Pinto (07.20.2007 @04:17:55 PM EDT)
• Didier Bizzarri (07.21.2007 @11:01:04 AM EDT)
• Daniel Bitin (07.21.2007 @11:48:54 AM EDT)
• Xianchao Xie (07.21.2007 @04:15:50 PM EDT)
• Jens Voz (07.23.2007 @09:37:48 AM EDT)
• David Tenne (07.23.2007 @02:32:44 PM EDT)
• Jesus Urresti Ibanez (07.25.2007 @09:11:49 AM EDT)
• Du Yang (07.25.2007 @06:09:28 PM EDT)
• Josep M. Pujol (07.26.2007 @05:50:33 PM EDT)
• Greg Janee (07.28.2007 @01:29:22 AM EDT)
• Steve Plate (07.30.2007 @01:30:43 AM EDT)
• Jeremy Mardon (07.30.2007 @07:11:12 PM EDT)
• Elmar Zander (07.30.2007 @08:01:53 PM EDT)
• Janos Nagy and Geza Radics (07.31.2007 @04:37:20 PM EDT)
• Adriano A. Batista (07.31.2007 @10:25:23 PM EDT)