I asked a question over on Math.SE, pointing to this puzzle, and @Elaqqad gave a very long answer with lots of links to math that might help. @Elaqqad also produced a concrete solution with only 59 tests — beating @noedne's long-standing solution of 63 tests!
The "wolves and sheep" puzzle is a specific case of non-adaptive group testing. ...
This is only solvable in the sense that no solution is better than another.
Since the judge has no prior knowledge, starting at any position is just as good, so the judge might as well toss a (38-sided) coin to decide, (treating the line as if its ends were connected to each other) and this is still optimal for the judge.
If the judge does that, it becomes ...
If there is exactly one imbalance $a<b$, then coin $a$ weighs 9 g and coin $b$ weighs 11 g.
If there are two imbalances $a<b$ and $c<d$, then weigh only coins $a$ and >!$c$ (one extra weighing)
Case 1: $a<c \implies$ coin $a$ weighs 9 g and coin $d$ weighs 11 g
Case 2: $c<a \implies$ coin $c$ weighs 9 g and coin $b$ weighs 11 g
The answer is
Unfortunately, this answer is neither elegant nor easy to explain since I found it via brute force. It's pretty disappointing to solve a puzzle this way, but I don't think anyone explained a correct answer yet (at least before I was sniped by Charles Gleason!).
The General Approach
Consider the case of 9 coins with one heavier than the rest....
Let's assume for the moment that the judge chooses the starting location at random uniformly, and similarly the direction of travel.
In that case, the location at either end of the row is the best for the Janets. Clearly every one of them is equally likely to be the first to be examined. The ones at the ends have only one neighbour, while all the rest have ...
With the assumption that you can distinguish the coins by either marking them or retrieving them in the reverse order by stacking them on the scale, then worst case is seven weighings:
Arrange the coins into a grid of 4 rows and 5 columns
Weigh Row1 against Row2
Weigh Row3 against Row4
At this point, you know either:
Which row has the heavy coin AND ...