IBM Research | Ponder This | April 2015 solutions

# Ponder This

## April 2015

<<March April May>>

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.

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