Poor Farmer Ned

Farmer Ned bought a pasture surrounded by a beautiful wooden fence. He built a dividing fence and moved his horses to one end and his cows to the other. But the next day, most of his new neighbors called to report animals in their yards. It turns out the fences were not as sturdy as he thought.

Ned repairs the fence, but the next day his neighbors call again. There are horses in Dan's yard and cows in Tim's.

Again Ned fixes the fence but the next day, Dan reports cows and horses trampling his garden.

Ned drives the animals back, but doesn't have time to do much for the fence. Sure enough, the phone rings the next morning. Gus, his neighbor to the north, reports horses in his yard. Lisa, his neighbor to the south, reports cows in her yard, and Dan is on the phone as well.

Ned fixes the fences, but the following day there are horses in Tim's lawn and cows in Dan's.

Grumpily, Ned surrounds the property with a steel wall. This mostly works, but the horses jump over it to visit Tim.

Ned makes that part of the wall higher, but the horses just escape out the opposite side. Meanwhile, the cows have escaped to visit Dan and Lisa.

Ned makes the wall higher and for good measure, electrifies it. The next morning his phone is silent. No animals have escaped!

That night he goes to sleep feeling content. But the next morning, the phone rings. Who is it?

• I'm guessing it's PETA because all the animals died. Commented Aug 23, 2023 at 17:50

It is

Dan, calling to complain about cows.

You can get an idea of the layout of Ned's farm and the surrounding farms from the description.
Ned divided the pasture in half, with cows at one end and horses at the other. It is stated that Gus lives to the north, and Lisa lives to the south, and they each only ever complain about one specific animal (Gus -> Horses and Lisa -> Cows), so we can assume the horses are at the north end, and the cows are at the south end:

Dan and Tim are on the left and right, but we're not sure which is which.

Each day, certain neighbours call to complain about certain animals, thus:

Day   Dan   Tim   Gus   Lisa
----------------------------
1   ( most - not specified)
2     H     C
3    CH
4     ?           H      C
5     C     H
6           H
7    CH                  C
8          ( none )


From this data, we can get a general idea of which fences were broken each day (shown in red). This assumes Dan is to the west and Tim is to the east. If the assumption is incorrect, then we just need mirror images of these figures:

The orange lines indicate options, since it was not specified which animal Dan complained about. Day 1 is not filled in, because the information is too vague.

Looking at the green lines, the pattern of the broken fences quickly becomes obvious, and we can fill in the missing data, which meshes with the vague descriptions given for the first and fourth days:

Thus

only one fence breaks on the 9th day; and it is the one connecting the cow pasture to Dan's farm.

• that graphical representation is perfect Commented Aug 24, 2023 at 9:38
• And on the tenth day, no neighbors complain, but poor Ned finds his cows and horses mixed together... Commented Aug 24, 2023 at 10:41
• We can even fit Day 0 into the pattern, when Ned had just the perimeter fence but no divider. @JeffZeitlin For Day 10 to fit, Tim and Dan need to start erecting their own fence parallel to Ned's, which is probably a reasonable course of action at that point. Commented Aug 24, 2023 at 14:52
• @NuclearHoagie - Tim/Dan doesn't have to build a fence for day 10; just look at the last digit alone - it effectively becomes a repeat of "day 0". Commented Aug 24, 2023 at 15:46