Ponder This Challenge - April 2015 - 7x7x7 with as much Peters

Ponder This Challenge:

A 7x7x7 cube is centered at the origin of a three-dimensional grid. The exact middle-point of each cell contains one character from the Peter Pan story. All the cells outside the cube are filled with Captain Hooks. Inside the cube some of the cells contain Peter Pans and the others contain Wendys.

The Captain Hook team wins if any one Hook can see at least one Peter Pan, but no Wendy, when looking in the direction of any one of his 26 immediate neighbors. We assume that all cell walls are transparent and you can see through them in any given direction.

Find a way to put at least 161 Peter Pans in the cube without letting Captain Hook win.

Supply your answer as a list of 343 characters: P for Peter Pan and W for Wendy. Start from the top close left corner and move left to right; top to bottom; close to far.

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

  A few years ago (https://www.research.ibm.com/haifa/ponderthis/challenges/July2007.html), we posted a challenge about building a fence to prevent viewing through a square property.

  This month's challenge is a three-dimensional discrete version of the same problem. The optimal solution for the two-dimensional discrete version is

  WWWPPW

  PPPPPW

  PPPPPW

  WPPPPP

  WPPPPP

  WPPWWW

  The trivial way to extend it to three dimensions is to have two opposite faces full of Wendies and the other four half filled. Tweaking a little, we get 170 Peter Pans:

  WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW

  WWWPPPWPPPPPPWPPPPPPWWPPPPPWWPPPPPPWPPPPPPWPPWWWW

  WWWPPPWPPPPPPWPPPPPPWWPPPPPWWPPPPPPWPPPPPPWPPWWWW

  WWWPPPWPPPPPPWPPPPPPWWPPPPPWWPPPPPPWPPPPPPWPPWWWW

  WWWPPPWPPPPPPWPPPPPPWWPPPPPWWPPPPPPWPPPPPPWPPWWWW

  WWWPPPWPPPPPPWPPPPPPWWPPPPPWWPPPPPPWPPPPPPWPPWWWW

  WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW

  The best solution that we know of has 172 Peters, which requires some Wendies in the inner 5x5x5 cube:

  WWWWWWWWWWWWWWWWWWWWWWWWPWWWWWWWWWWWWWWWWWWWWWWWW

  WWWWPPWPPPPPPWPPPPPPWPPWPPPPWPPPPPPWPPPPPPWPPWWWW

  WWPWWPWPPPPPPWWPPPPPPPPPPPPWPPPPPPWWPPPPPPWPWWPWW

  WWWWPPWPPPPPPWPPPPPPWPPPPPPWWPPPPPPWPPPPPPWPPWWWW

  WWPWWPWPPPPPPWWPPPPPPPWPPPPPPPPPPPWWPPPPPPWPWWPWW

  WWWWPPWPPPPPPWPPPPPPWPPPWPPPWPPPPPPWPPPPPPWPPWWWW

  WWWWWWWWWWWWWWWWWWWWWWWWPWWWWWWWWWWWWWWWWWWWWWWWW

  And thanks to LaurV for the interesting discussion on the Mersenne forum http://www.mersenneforum.org/showthread.php?t=20151.

Solvers

• James Dow Allen (04/01/2015 03:07 AM EDT)
• Lorenz Reichel (04/01/2015 04:05 AM EDT)
• Kang Jin Cho (04/02/2015 02:33 AM EDT)
• *Don Dodson & David Dodson (04/02/2015 05:15 PM EDT)
• Saverio Maresca (04/03/2015 03:17 AM EDT)
• Dan Dima (04/03/2015 05:05 AM EDT)
• Chuck Carroll (04/03/2015 06:54 AM EDT)
• Graham Hemsley (04/03/2015 06:59 AM EDT)
• Niketh Nair (04/03/2015 05:54 PM EDT)
• *Robert Gerbicz (04/03/2015 08:10 PM EDT)
• *Ariel Landau (04/04/2015 12:10 PM EDT)
• Florian Fischer (04/04/2015 06:17 PM EDT)
• Sean Egan (04/04/2015 09:21 PM EDT)
• *Eli Biham (04/05/2015 10:00 AM EDT)
• Jim Clare (04/05/2015 12:15 PM EDT)
• Benjamin Phillabaum (04/05/2015 12:44 PM EDT)
• *Motty Porat (04/05/2015 07:36 PM EDT)
• Shirish Chinchalkar (04/05/2015 11:04 PM EDT)
• LaurV (04/06/2015 08:02 AM EDT)
• Harald Bögeholz (04/07/2015 03:13 PM EDT)
• *Todd Will (04/07/2015 03:40 PM EDT)
• Andreas Stiller (04/07/2015 06:21 PM EDT)
• Piotr Nadczuk (04/08/2015 07:15 AM EDT)
• Stefan Rüdrich (04/08/2015 11:13 AM EDT)
• *Alex Wagner (04/09/2015 01:33 AM EDT)
• John Tromp (04/09/2015 09:22 PM EDT)
• Ricardo Koller (04/13/2015 12:30 AM EDT)
• Joseph DeVincentis (04/13/2015 06:01 PM EDT)
• Chintan Maggu (04/13/2015 11:15 PM EDT)
• Zhizhong Zhou (04/14/2015 11:30 AM EDT)
• *Emilio Del Tessandoro (04/14/2015 05:02 PM EDT)
• Mohammed Koni (04/14/2015 07:03 PM EDT)
• Peter Gerritson (04/15/2015 08:13 AM EDT)
• *Ryan Ersland (04/16/2015 06:19 PM EDT)
• *Dafni Harel (04/18/2015 10:26 AM EDT)
• Daniel Bitin (04/19/2015 10:03 AM EDT)
• *David Greer (04/19/2015 10:38 PM EDT)
• Shouky Dan & Tamir Ganor (04/20/2015 03:25 PM EDT)
• *Armin Krauss (04/21/2015 04:48 PM EDT)
• Liubing Yu (04/23/2015 11:02 AM EDT)
• Frank Methling (04/23/2015 02:15 PM EDT)
• Jan Fricke (04/24/2015 04:41 PM EDT)
• *Radu-Alexandru Todor (04/25/2015 06:56 AM EDT)
• *Mark Pervovskiy (04/25/2015 10:55 AM EDT)
• Tobias Zurell (04/26/2015 01:59 PM EDT)
• Rajendran Thirupugalsamy (04/27/2015 12:21 AM EDT)
• Stéphane Higueret (04/28/2015 05:24 PM EDT)
• Suriya Subramanian (04/29/2015 08:12 AM EDT)
• Aviv Nisgav (04/29/2015 11:25 AM EDT)
• *Cadence Marseille (04/29/2015 12:12 PM EDT)
• *Muralidhar Seshadri (04/29/2015 05:46 PM EDT)
• *Stéphane Higueret (04/30/2015 12:17 AM EDT)
• River Misaki (04/30/2015 07:42 AM EDT)
• Erik Wünstel (04/30/2015 03:34 PM EDT)
• Elendia (04/30/2015 03:42 PM EDT)