When I was about 14 (and in the advanced math set) my math teacher gave me this problem. Specifically the one with the goat in a square field tethered in the ce

When I was about 14 (and in the advanced math set) my math teacher gave me this problem. Specifically the one with the goat in a square field tethered in the center of one side. I spent ages trying to solve it and managed to convince myself that it didn't have an analytical solution - well at least not one I could come up with. So I invented an iterative process to binary search the solution on my calculator. I came up with an answer in the end to 5 or so significant figures which I was happy with. That was the first math problem that I met that didn't have an analytical solution but I could solve so it was one of those breakthrough moments in my maths learning. My math teacher was happy with the solution. She'd given it to us deliberately as an example of a problem with a solution, but no analytical solution as a kind of trick question in order to make us think. Great teacher!