The goat problem, first formulated in 1894, asks one to calculate the length of a tether by which a goat is attached to the inner circumference of a circular field of unit area, allows the goat to graze exactly one half of the area of the field.

An equation for the radius r of the tether can be obtained by relatively straightforward trigonometry as:

This is a transcendental equation, which has no closed-form solution, and thus r must be approximated by iteration. Performing this computation yields the approximation of 1.15872847â€¦.

The goat problem has long been used an example of a problem which is simple to state, straightforward to solve, yet whose exact solution cannot be written out in terms of simple mathematical operations and functions and must be solved numerically. But in 2020, Ingo Ullish published â€śA Closed-Form Solution to the Geometric Goat Problemâ€ť (behind Springer paywall, jailbreak on Sci-Hub here), in which he used complex analysis to obtain the following closed-form solution as the ratio of two contour integrals in the complex plane:

Now, some may quibble whether this is a closed-form solution since, in practice, the contour integrals themselves must be evaluated iteratively, most efficiently by a fast Fourier transform.

Interestingly, the goat problem generalised to dimensions greater than 2 is *simpler*, and has closed-form solutions which, as the number of dimensions goes to infinity, converge to r=\sqrt{2}.