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?

ANSWERS

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

Circular Ponds
n=1

r = 1/2 = .500 		n=2

r = 5/8 = .625 		n=3

r = 1/√2 = .707+ 		n=4

r = 5√377/128 = .758+ 		n=5

r = .808+ (JT)
n=6

r = .844+ (JT) 		n=7

r = .869+ (MM) 		n=8

r = .900+ (JH) 		n=9

r = .955+ (JH)
n=10

r = .976+ (JH) 		n=11

r = 1.020+ (RE) 		n=12

r = 1.039+ (JH)

Triangular Ponds
n=1

s = 3/2 = 1.500 		n=2

s = 1+2/√3 = 2.154+ 		n=3

s = 2.288+ (MM)
n=4

s = 2.443+ (JH) 		n=9

s = 3.013+ (JH)
n=10

s = 3.116+ (JH) 		n=11

s = 3.195+ (JH)

Square Ponds
n=1

s = 1 		n=2

s = 1.192+ (MM) 		n=3

s = √2 = 1.414+ 		n=4

s = 1.480+ (MM)
n=5

s = 1.531+ (JH) 		n=6

s = 1.620+ (JH) 		n=7

s = 1+1/√2 = 1.707+ 		n=8

s = 1.754+ (JH)
n=12

s = 1.971+ (JH) 		n=14

s = 2.076+ (JH) 		n=19

s = 2.254+ (JH)

Pentagonal Ponds
n=1

s = (5–√5)/4 = .690+ (JH) 		n=2

s = .885+ (JH) 		n=3

s = .969+ (JH) 		n=4

s = 1.093+ (JH)

Hexagonal Ponds
n=1

s = 1/√3 = .577+ (MM) 		n=2

s = (3+2√3)/9 = .718+ (MM) 		n=3

s = .790+ (MM)
n=4

s = .830+ (JH) 		n=7

s = 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.