# Problem of the Month (August 2019)

Suppose you have a circular pond that you want to reach the center of, and all you have is a bunch of 1-yard-long boards that you can overlap. what is the largest pond that you can reach the center of using n boards? What if the pond is not circular, but square or triangular?

Solutions were sent by Maurizio Morandi, Jean Hoffman, Jeremy Tan, and Richard Erikson.

Circular Ponds
 n=1r = 1/2 = .500 n=2r = 5/8 = .625 n=3r = 1/√2 = .707+ n=4r = 5√377/128 = .758+ n=5r = .808+ (JT)
 n=6r = .844+ (JT) n=7r = .869+ (MM) n=8r = .900+ (JH) n=9r = .955+ (JH)
 n=10r = .976+ (JH) n=11r = 1.020+ (RE) n=12r = 1.039+ (JH)

Triangular Ponds
 n=1s = 3/2 = 1.500 n=2s = 1+2/√3 = 2.154+ n=3s = 2.288+ (MM)
 n=4s = 2.443+ (JH) n=9s = 3.013+ (JH)
 n=10s = 3.116+ (JH) n=11s = 3.195+ (JH)

Square Ponds
 n=1s = 1 n=2s = 1.192+ (MM) n=3s = √2 = 1.414+ n=4s = 1.480+ (MM)
 n=5s = 1.531+ (JH) n=6s = 1.620+ (JH) n=7s = 1+1/√2 = 1.707+ n=8s = 1.754+ (JH)
 n=12s = 1.971+ (JH) n=14s = 2.076+ (JH) n=19s = 2.254+ (JH)

Pentagonal Ponds
 n=1s = (5–√5)/4 = .690+ (JH) n=2s = .885+ (JH) n=3s = .969+ (JH) n=4s = 1.093+ (JH)

Hexagonal Ponds
 n=1s = 1/√3 = .577+ (MM) n=2s = (3+2√3)/9 = .718+ (MM) n=3s = .790+ (MM)
 n=4s = .830+ (JH) n=7s = 1.017+ (JH)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 8/26/19.