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# Ponder This

## July 2007

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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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