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A Centuries-Old Math Problem Involving Goats Finally Has An Exact Solution

Here is a question that sounds simple at first, but is actually quite difficult to solve:

If you tie a goat to a circular fence, how long does the rope have to be for the goat to have access to half the area?

The question is more than 270 years old and we have just now solved it.

The problem was designed as a math puzzle for women.

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A variation of this problem first appeared in The Ladies Diary: Or, The Woman's Almanack. It was the first English publication designed specifically for women. The almanac aimed to encourage women to engage in mathematics and demonstrate their aptitude for math. It was very popular and was published for over 100 years (1704 to 1840).

Why a goat?

Oddly enough, the original problem featured a horse. No one knows why it was changed to a goat. The problem was also a little different. It concentrated on the outside of the circle instead of the inside:

If a horse is tied to a circular fence with a rope the length of the circumference of the fence, how much grass can it graze on?

Exterior puzzles are a little easier to solve.

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The original puzzle was solved using trial and error and a table of logarithms. The final answer was just an approximation though. But more importantly, the puzzle inspired many other variations of a similar question, including the goat question of today.

There are ways to calculate an approximate answer.

But, until now, there wasn't a mathematical formula that produced the exact answer. Ingo Ullisch determined the solution by reducing the problem to a single transcendental equation. Then he approached the problem using complex analysis, which is a branch of mathematics that applies analytics tools to expressions with complex numbers. His formula still produces a number which is of little practical use, but it is about the journey, not the destination... sometimes.

This answer will not revolutionize math.

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The problem is very specific and has little application. But, looking at novel problems can create new solutions with approaches that may be applied to other problems. So, who knows, maybe this exercise in animal bondage will lead to a bigger breakthrough later.

h/t: Quanta

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