Educational Mathematics – A Bewitching Barnyard Task

Posted Feb 18, 2021

It has happened to all of us. We work on a mathematics problem, believing that we’ve found a beautiful and elegant solution, only to find a snag somewhere along the chain of reasoning.  The same is true among the best mathematicians, even when tackling seemingly simple problems. Consider the following exercise, first posited over 270 years ago: 

Imagine a circular fence that encloses one acre of grass. If you tie a goat to the inside of the fence, how long of a rope do you need to allow the animal access to exactly half an acre? 

Grab a pencil and a piece of paper. Open up a dynamic geometry app. Come back after pondering awhile (perhaps reaching an apparent solution). 

If you’ve landed on an answer, it is likely to be an approximation at best. Many mathematicians have tackled this problem, thinking that they have finally captured an elusive solution, only to be thwarted by slipping into rounded decimal methods.   

In fact, “nobody knows an exact answer to the basic original problem,” according to Mark Meyerson, an emeritus mathematician from the U.S. Naval Academy. Although it is believed that in early 2020, Ingo Ullisch, a German mathematician, may have accomplished some progress on finding an exact solution, his work entails transcendental equations, complex analysis, and contour integral expressions. This is definitely not what one expects from such a surface-simple problem. 

To learn more about the twisty path to a solution and the history of this “low threshold, high ceiling” application of mathematics, read more here.

– Jeff Smith, MAEM Graduate Director

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