Trying to solve this puzzle as usually interpreted seems futile. Many have tried, even with computer programs, with the result of "it's impossible". Unless the two claimed solvers come up with something to back their claims, or we can think of new ways to interpret it, this puzzle seems like a waste of time.

I've tried a whole bunch of variations with my program, no results. These are the variables:

- Defining the value of a concrete piece of information like "N plus V is 112". A student would consider the given piece of information to help iff given it, { (A: there would be a single answer), or (B: some of the currently possible answers would be eliminated) }.
- Defining how the students would answer to "would it help if I told you X", with X like "N plus V" for example. A student would say yes, iff { (A: any concrete variant of X), (B: all concrete variants of X), or (C: the actual concrete variant of X [by "magical knowledge"]) } would help according to the definition chosen above.
- Defining how the numbers are "labeled". If F is indistinguishable from N and/or V, it is given as a number 1..5. { (A: all three indistinguishable), (B: N marked, but F indistinguishable from V), (C: F marked [or not expressed as a number], but N indistinguishable from V), (D: V marked, but N indistinguishable from F), or (E: all three marked, or each recipient otherwise knows which one they got) }.
- Stretching the definition of a three-digit number to allow zeroes. The possible values for N are the otherwise allowed subset of { (A: 111..999), (B: 000..999), or (C: 000..999 where LCM(N) is defined, as defined below) }.
- Defining LCM(N) if N includes zeroes. If LCM(N) is undefined, F simply can't be LCM with that N. { (A: undefined), (B: LCM(000) = 0; else undefined), (C: LCM of the non-zero digits; LCM(000) = 0), or (D: LCM of the non-zero digits; LCM(000) is undefined) }.
- Defining how the inclusion of zeroes effects the "labeling". { (A: the leading zeroes in N are printed, i.e. if N has leading zeroes, N is recognizable from F and V), or (B: the leading zeroes in N are omitted, i.e. the regular labeling applies and N may be indistinguishable) }.
- Altering the digit relation rules of N, guessing possible error in the puzzle with the complicated expression. With N="xyz", i.e. 100x + 10y + z (with x y z in 0..9), x y z are restricted by: { (A: x<=y<=z), (B: x<y<z), (C: x>=y>=z), or (D: x>y>z) }.
- Adding a constraint for V. { (A: the puzzle stands as stated), (B: V must be a three-digit number), or (C: V must satisfy the same constraints as N) }.

I've examined almost all combinations of the above. The excluded ones combine 2C (the magical knowledge) with stretching the puzzle definition by more than one of (not 4A), (not 7A), (not 8A).

There is one combination that does produce some answers: 1A, 2C, 3A/C, 4A, 7A, 8B. 3A and 3C are equivalent because with 8B, N and V are distinguishable from F anyway. Out of 270 total possibilities, there are 6 working answers to the puzzle. 2C produces multiple answers in many combinations I excluded as well. Of the excluded more exotic combinations, some do have an unique answer, but truth be told, I don't like 2C one bit, and combining it with many obvious distortions of the puzzle just goes too far.

Of course it's always possible that I have a bug somewhere. I'd always welcome someone checking my results. Else we need even more stretching of the puzzle interpretation than I have applied. I may have missed something much more likely than my weirder scenarios. Any ideas?

If someone writes the same kind of program to verify my work, it would be useful to compare some of the intermediate numbers. That would work towards verifying both programs, as the same overall result of no working combinations doesn't tell all that much. PM me if you'd like some data to compare to.

I just had an interesting thought. In all of my calculations I've assumed that all the rules are common knowledge. That doesn't necessarily have to be true. For example, what if the students don't know which of the number labeling schemes is in use? The most likely example: F is given as the function and not a number, but the receivers of N and V don't know that. This is a bit harder to simulate, I will have to think about it. It would be one evil puzzle, and quite unlikely, but the analysis just isn't comprehensive without considering the possibility.

Another extension I thought about: Maybe 1 and 2 need to be chosen independently for the different questions. That would be weird if it's intentional and just forgotten out of the puzzle, but I think it's well possible that the author consistently made this mistake when checking their solution. "If N/V was odd or even" feels quite different from "the sum of N and V" after all. I'll add this possibility when I next get into this.

Edit: No solutions with the last addition, although I had to exclude more of the weird combinations when there are some Cs in 2, to limit the run time.