I've seen a couple of places explain that the answer is C because it's mentioned the most times, but I don't understand the logic behind that. My bigger problem is that I can't work backward from the answer, so I assume I don't even understand the question.
Here's how I parse it: (I'm going to use capital letters when referring to the reports and lowercase when referring to the modules themselves.) Each module's report lists all the fully operational ("on", for convenience) modules in the system; e.g., A says a, f, e, and b are on, and c and d are off. I tried checking for contradictions. Like if A says e is on but E says a is off, A or E must have an error. But they all contradict each other. Ultimately I settled on visualizing the reports as a matrix where the columns are the modules and the rows are which report said the modules were on. (1 is on, 0 is off, of course.)
a b c d e f
A 1 1 0 0 1 1
B 0 1 1 1 1 0
C 1 1 1 0 1 0
D 1 0 1 1 0 1
E 0 0 1 1 1 1
F 1 0 1 1 0 1
I thought at first that the identical reports, D and F, would have to be the accurate ones. They're both reporting on the same system, so wouldn't it make sense that they both give the same report? But if we assume they are accurate, the requirement that all the other reports have one error doesn't work out; A, B, and C have two errors.
c does stand out as being the only one that's mentioned in five reports, but what does it mean? That c is probably on? So what? With four modules on and only two accurate reports, I don't see why we'd have to assume that a module has to be on to give an accurate report. And even if we know from how often it's mentioned that C is an accurate report, where is the other accurate report? Shouldn't it be mentioned just as often?
So I tried counting errors and ended up with this matrix:
A B C D E F
A 0 2 1 2 2 2
B 2 0 1 2 1 2
C 1 1 0 2 2 2
D 2 2 2 0 1 0
E 2 1 2 1 0 1
F 2 2 2 0 1 0
If the rows (or columns; it's symmetric) are the reports we assume are accurate and the columns (or rows) are the number of errors when compared with the given assumed-accurate report, the two accurate reports should have a row (or column) of four ones and two zeros. I think.
And even if any of that worked, I don't see how, once you find the accurate reports, you decide which one is the unstable module.
At this point I can only assume either none of the programmers involved noticed there's no solution or I'm doing something wrong. Occam says I'm doing it wrong.
I thought I might've misinterpreted the question and each module reported only three modules as operational, so I redid it like that and got nowhere.