I first misread "goat tied to the inside of the fence" as meaning tied to a post in the center of the circle, and was confused at why everyone thought this question was so hard.
I initially assumed that the goat was tied to the fence in such a way that the non-goat rope-end could move completely around the circumference of the fence and not be fixed to a single point.
I was also at first confused, and I find the problem as published in 1894 (i.e. almost 130 years ago) much clearer:
"A circle containing one acre is cut by another whose
centre is on the circumference of the given circle, and
the area common to both is one-half acre. Find that
radius of the cutting circle."
I think that formulation had to be at least mentioned near the beginning of the text, when not used in the first sentence.
> Your goat eats the "donut" and GP's goat eats the "hole", each eating half of the area.
Assuming the two goats are on equally long leashes, this doesn't come close to being true. They each eat circular arcs which run through half a radius. But the outside goat's circles are huge and the inside goat's circles are tiny; the inside goat covers 25% of the area.
I was just drawing what you said on the whiteboard, and I was like.. uh, am I missing something or am I a super genius? I am positive the latter is incorrect.
I had to re-read the first paragraph multiple times until I understood the goat was not tethered to the center.
