So the cow has enough rope to go all the way around the circumference of the silo, right?

I’m not so great at creating formulas, but it looks to me that as the cow moves from the left side (anchor point) to the right side of the silo (opposite point to the anchor), the radius of his “freedom of movement” circle goes from pi to pi/2.

I would attempt to graph and/or formulamajize what that would look like but it’s late at night for me >.<

Plus, you’ll have an interesting answer every time someone asks why that fence is there! That’ll be what makes it worth it, right?

What if the silo was a polygon (with the same perimeter) instead of a circle? Here’s something you could try: Start with a line (area becomes a complete circle), then a triangle, square, pentagon, et cetera, and see what the area approaches as the number of sides approaches infinity. (I envy anyone with sufficient free time to actually try this approach, but at least I think it will deliver an answer… eventually. Maybe.)

I did think it was a strange coincidence that palindromes showed up in the list. Did you make this puzzle yourself? If so, how did you notice this pattern? :D

I may have cheated because I did it programmatically with PHP :P
1) Made an array of all the palindromes under 1000 (1…9, 11…99, 101…999)
2) Made an array of all the sums of these palindromes
3) Made an array of all the numbers from 100 to 999 which are not in the array from step 2