A recent USAMO problem asked the contestant to prove that
is an integer for every . Unfortunately, it appears that this is a special case of the so-called hook-length formula, applied to a Young tableau, and several students appealed to this fact without proof to produce one-line solutions lacking any substance. This has led to a major controversy about how such solutions should be graded, in particular whether they should receive the treatment for “essentially correct solutions”, or the treatment for “essentially not solved”.
In this post I want to argue that I think that these solutions deserve a score of .
However, before I do so, I would like to make some disclaimers:
- This issue is apparently extremely polarized: everyone seems to strongly believe one side or the other.
- This was an extremely poor choice of a USAMO problem, and so there is no “good” way to grade the HL solutions, only a least bad way. The correct solution to the dilemma is to not have used the problem at all. Yet here’s the bloodied patient, and here we are in the emergency room.
- While I am a grader for the USAMO, I am one of many graders, and what I say in this post does not necessarily reflect the viewpoints of other USAMO graders or what score the problem will actually receive. In other words, this is my own view and not official in any way.
One last remark is that I do not consider the hook-length formula to be a “well-known” result like so many contestants seem to want to pretend it is. However, this results in the danger of what constitutes “well-known” or not. So in what follows I’ll pretend that the HL formula is about as well-known as, say, the Pascal or Zsigmondy theorem, even though I personally don’t think that this is the case.
One final disclaimer: I am unlikely to respond to further comments about an issue this polarized, since I have already spent many hours doing so, and I’ve done enough of my duty. So if I don’t respond to a comment of yours, please don’t take it personally.
2. Rule for citations
Here is the policy I use for citations when grading:
- You can cite any named result as long as it does not trivialize the problem.
- If the result trivializes the problem, then you are required to prove the result (or otherwise demonstrate you understand the proof) in order to use it.
This is what I’ve heard every time I have asked or answered this question; I have never heard anything to the contrary.
Some people apparently want to nit-pick about how “trivialize” is not objective. I think this is silly. If you really want a definition of “trivialize”, you can take “equivalent to the problem or a generalization of the problem” as a rule of thumb.
Clearly it follows from my rule above that the hook-length formula deserves grading, so the remainder of the post is dedicated to justifying why I think this is the correct rule.
3. Subjective grading
I would rather have an accurate subjective criteria than a poor objective one.
In an ideal world, grading would be completely objective: a solution which solves the problem earns points and a solution which does not solve the problem earns points. But in practice this is of course not possible, unless we expect our contestants to write their solutions in a formal language like Coq. Since this is totally infeasible, we instead use informal proofs: students write their solutions in English and human graders assign a score based on whether the solution could in principle be compiled into a formal language proof.
What this means is that in grading, there are subjective decisions made all of the time. A good example of this is omitting steps. Suppose a student writes “case is similar [to case ]”. Then the grader has to decide whether or not the student actually knows that the other case is similar or not, or is just bluffing. One one extreme, if and really are identical, then the grader would probably accept the claim. On the other extreme, if and have some substantial differences, then the grader would almost certainly reject the claim. This implies that there is some intermediate “gray area” which cannot be objectively defined.
Fortunately, most of these decisions are clear, thus USAMO grading appears externally to be mostly objective. In other words, accurate grading and objective grading are correlated (yay math!), but they are not exactly the same. All scores are ultimately discretion by human graders, and not some set of rigid guidelines.
Citations are a special case of this. By citing a theorem, a student is taking advantage of convention that a well-known proof to both the student and grader can be omitted from the write-up.
4. Citing the problem
In light of this I don’t think you should get points for citing the problem. I think we all agree that if a student writes “this is a special case of IMO Shortlist 1999 G8”, they shouldn’t get very many points.
The issue with citing HL in lieu of solving the problem is that the hook-length formula is very hard to prove, and it is not reasonable to do so in an olympiad setting. Consequently, it is near certain that these students have essentially zero understanding of the solution to the problem; they have not solved it, they have only named it.
Similarly, I think you should not earn points for trivializing a problem using Dirichlet, Prime Number Theorem, Zsigmondy, etc. This is also historically consistent with the way that grading has been done in the past (hi Palmer!).
5. Citing intermediate steps
Now consider the usage of difficult theorems such as Dirichlet, Prime Number Theorem, etc. on a solution in which they are merely an intermediate step rather than the entire problem. The consensus here is that citing these results is okay (though it is not unanimous; some super harsh graders also want to take off points here, but they are very few in my experience).
I think this is acceptable, because in this case the contestant has done 90% of the solution, and cannot do the remaining 10% but recognize it as well-known. In my eyes this is considered as essentially solving the problem, because the last missing bit is a standard fact. But somehow if you cannot do 100% of the problem, I don’t think that counts as solving the problem.
What I mean is there is a subjective dependence both on how much of the solution the student actually understands, and how accessible the result is. Unfortunately, the HL solutions earn the worst possible rank in both of the categories: the student understands 0% of the solution and moreover the result is completely inaccessible.
6. Common complaints
Here are the various complaints that people have made to me.
- “HL is well-known.”
Well, I don’t think it is, but in any case that’s not the point; you cannot cite a result which is (a) more general than the problem, and (b) to which you don’t understand the proof.
- “Your criteria is subjective!”
So what? I would rather have an accurate subjective criteria than a poor objective one.
- “It’s the problem writer’s fault, so students should get .”
This is not an ethics issue. It is not appropriate to award points of sympathy on a serious contest such as the USAMO; score inflation is not going to help anyone.
- “It’s elitist for the graders to decide what counts as trivialized.”
That’s the grader’s job. Again I would rather have an accurate subjective criteria than a poor objective one. In practice I think the graders are very reasonable about decisions.
- “I don’t think anyone disputes that the HL solution is a correct one, so certainly the math dictates a .”
I dispute it: I don’t think citing HL is a solution at all.
- “Why do we let students use Pascal / Cauchy / etc?”
Because these results are much more reasonable to prove, and the “one-line” solutions using Pascal and Cauchy are not completely trivial; the length of a solution is not the same thing as its difficulty. Of course this is ultimately subjective, but I would rather have an accurate subjective criteria than a poor objective one.
- “HL solutions have substance, because we made an observation that the quantity is actually the number of ways to do something.”
That’s why I wish to award instead of .
- “Your rule isn’t written anywhere.”
Unfortunately, none of the rules governing USAMO are written anywhere. I agree this is bad and the USAMO should publish some more clear description of the grading mechanics.
- “The proof of the HLF isn’t even that complicated.”
Are you joking me?
In summary, I don’t think it is appropriate to give full marks to a student who cannot solve a problem just because they can name it.