For more than 270 years, scientists and enthusiasts of mathematics have been trying to find the exact solution of the problem of how to determine the length of the rope that allows the goat to graze on half the Acre of the Earth if it is attached to the inside of the round fence. The task seems simple and may seem familiar from the school geometry course, but in reality it turned out to be so complicated that throughout this time it was possible to get only close answers. However, this year, the German mathematician Ingo Ullish proposed the first accurate solution. This puzzle, solving the problem that occupied the minds of mathematicians for more than two centuries.
The origins of the problem: from horses to goats
The first references to such tasks can be found in the old London magazine The Ladies Diary, published in 1748. In that task, it was necessary to determine which area a horse could graze, tied outside to a round fence if the length of the rope is equal to the length of the fence circumference. This option was called “external task”, since the horse was outside the circle. The decision, then proposed by one of the readers of the magazine, was only close, but at that time it seemed an acceptable result.
A one and a half centuries, in 1894, the task returned to the mathematical arena in the magazine “American Mathematical Monthly”, but in a more complex form: now the animal, in this case, the goat was tied inside a round fence, and it was necessary to determine the length of the rope so that the goat could graze exactly half the area of the fenced circle. This option was called the “internal problem” and turned out to be much more difficult to solve. Unlike an external task, where based on the length of the rope it was possible to calculate the area, now it was necessary to find the length of the rope, based on a predetermined area.
Mathematical difficulties and multidimensional approaches
For decades, the task of a goat has repeatedly rethought, and various mathematical magazines proposed their versions of its solution. Some mathematicians tried to solve the problem using various forms of fences, such as ellipses or squares, instead of a circle. In the 1960s in the literature on this topic, the animal was suddenly replaced by a goat, and this image was fixed as the