Note that you must check your institution's policy before questioning the student. There may be due process requirements about a formal accusation of cheating and provision of representation first. In the worst case, this could lead to expensive lawsuits.

@user71659, the beauty of this solution is that it really doesn't have to be anything other than an enthusiastic interview of a genius. Surely there is no due process for formal accusation of being a genius? :) There's a distinction between being disingenuous, and factually not knowing and simply recognizing that you don't know and shouldn't assume.

The problem is that let's say the student does admit to cheating or else provides strong evidence that they were. If you did not follow proper procedures, the admission may be considered inadmissible and you wouldn't be able to do anything. No different from a police officer interviewing a suspect. (In my current institution, you can't take any disciplinary action without going through the Dean of Students, who will review all evidence.)

@user71659, personally, I see that as an advantage, not a drawback. I'd rather forgive a student and tell him never to do it again, than to have him "get away" with it. And I'm a believer in second chances. (You may call me strange for that if you wish.)

@user71659 Police officers can interview any suspect for any reason and anything they say is admissible. Things like reasonable suspicion, probable cause, a warrant, a Miranda warning, and the like are only needed for custodial interrogations (where a reasonable person would not thing they were free to leave), if they know the person is represented by an attorney on this particular matter, or a few other very limited cases. At least in the United States. You can have a friendly chat even if you have an ulterior motive and that doesn't violate anyone's rights if you don't pressure them.

How does this answer not condition the student to expect a freebie for committing a serious infraction?

@djechlin, how does simply letting it slide not condition the student to expect cheating not to matter because no one ever even noticed?

@djechlin, more to the point: at this time you don't know that he cheated. You really honestly don't. Neither does the OP. Read Captain Emacs's comments, assume that this student is in the same situation, and then reread this answer.

This answer seems to suggest that either he is a genius or he cheated, I find that very problematic. I wouldn't say being the best in 80 in one single exam makes someone "a genuine genius", there's a hundred reasons why someone might solve a problem other's couldn't. If they don't seem like "a genius" in the interview, are really you just going to deduce that they cheated?

@JiK, I considered it obvious that there is a wide spectrum in between the self-demonstrating extremes. On the one hand the person may display a total lack of even basic understanding of the answer that they wrote down. On the other hand they may astonish you with new insights that you, the instructor, never even considered. Or it may be anywhere in between. It should be extraordinarily self-evident that this is an infinite gradient scale, not a binary choice—as are most things in live decision-making.

@Wildcard Well, the tone of this answer makes it look otherwise. The most likely outcome (quite clearly, I would say) is that he just happened to remember something that others didn't, which helped solve the problem, and that's it, but the answer doesn't even mention that possibility.

@DavidSchwartz The question isn't about police officers, so whatever police officers can and cannot do is irrelevant.

You absolutely must not interview the student without first checking your institutional policies. What if, for example, the other students argue that interviewing this candidate allowed them to better explain their answers to you, resulting in higher marks? Why weren't the other students given this opportunity?

"how does simply letting it slide not condition the student to expect cheating not to matter because no one ever even noticed?" I'm asserting that catching the student and letting him go is a significantly stronger signal that cheating is acceptable than failing to catch the student in the first place. The student goes from having to worry about not getting caught, to having to worry about not getting caught twice.

Of course we don't know. Your defense of your answer is skipping between "if he did" and "we don't know if he did." Assume my previous comment is scoped to the situation in which he turns out to have cheated, the problems with getting the student, assuming the student cheated, to fess up, notwithstanding.

I solved the exam problem by using my brain.

@djechlin, I understand your assertion. See my first two comments. It's clear we disagree on this point. I don't wish to argue it further. I will only add, in case it helps you understand my attitude, that there is a gradient scale of increasing disciplinary action, and that speaking to someone about the unacceptability of their actions (without any tangible penalties but with the inference that further trangressions may be dealt with harshly) is on that scale, and is different from "forgiving" as a "scot-free" idea. I meant the former though I used the word "forgive" in my comment.

Don't do this, really. Professors should not have the right to do this unless they actually have evidence to go on. Fishing for it, or intimidating students, is inappropriate. Hopefully, institutional policy/practice prevents such behavior - and if it doesn't, ethics should. Plus, OP is in a rather tenuous position since the course staff botched the exam.

According to the OP, the student knew (or looked up) a definition that others did not know. He did not crack anything unsolvable. Therefore, I don't see how this suggestion is helpful. The student will just say: "I had the definition memorized"...

"But if he can't explain the answer or how he arrived at it in any way, shape, or form, you will know he was cheating. And he will know that you know." This is wrong. I've forgotten many solutions, especially ones on exams where I was stressed etc. Hell, sometimes I look back at semi old solutions I wrote and can't follow at all! Just because they can't explain right then and there doesn't mean that they didn't do it. I can solve problems better on paper than explaining on a board.

@mmmmmmm "How'd you calculate this derivative?" "Ummm, haha, what's a derivative again cause I forget...and is this 'd' for diameter of x?"

@Wildcard: I once got 4 of 5 questions on a test right that supposedly depended on a concept I had no clue about. There were enough clues that if you solved them as the group they were you could completely not know the concept and still get it. My instructor couldn't figure how how I did it either. I obviously didn't cheat because I couldn't solve the one that really depended on it.

@einpoklum I heavily agree. We recommend that people "don't talk to the police" for very good reasons. If professors started doing this, then we would have to recommend students to "don't talk to professors." This would seriously impact the educational mission of the university.

"But if he can't explain the answer or how he arrived at it in any way, shape, or form, you will know he was cheating." --> This is plain wrong. It's a strong indicator, but it's not proof. Someone may have had a bad night and may absolutely not be in "Course X"-mode when spontaneously being asked such question. Anxiety or suprisedness in that situation may further block the brain. The summoned situation has nothing to do with the exam. And as unfair as an exam already is, this doubles the unfairness w.r.t. proper configuration: mood, sleep, mode, illnesses, medication, situation style..

... I know people who compete with Einstein on some day, just to fail trivial situations like counting pennies at the supermarket on the other day. And often enough, if we're honest, certain outcomes of exames are the result of lucky or bad configurations. It's only the total picture, not the single exam, that may tell something about a person's potency. tl;dr: Please don't confuse proofs with indicators. (P.S.: Anecdote: Ayrton Senna, multiple times Formula 1 champion, sometimes attributed his victories to the presence of "God" and couldn't further explain. What if the student does :D)

I love this answer. But to kick it up to the next level, praise him in class for being the only one to solve this question, then hand him the chalk/marker and let him explain it to the entire class. If he did cheat, it will become obvious then.

@Jennifer That's a cheeky one, I like that!

@Jennifer Individual students' marks are confidential. Announcing to the class that a certain student scored highly is inappropriate and may even be illegal.

@DavidRicherby academia is highly variable. Jennifer's suggestion would be fully appropriate in my school. In fact, IIRC, I've seen this happen, though the intent wasn't to catch a cheater, but to demonstrate the right solution and/or commend a student.