Unfair

I was a math prodigy. But everything changed when I encountered an unfair problem more like the real world than mathematics.

Math had always come easily to me. I majored in math at Penn State, and everyone in the department knew I was a student with special abilities.

Math sort of shifts on you. In high school and first college courses, math is about problem solving: find x, determine the angle, compute the integral, get the answer. There are a lot of mathematical students who are good problem solvers. Many of them go on to become excellent engineers.

Once you get into junior and senior level classes and graduate classes, the objective changes. Your job is not to solve problems, it is to prove things. You are seeking to find general statements of truth. You must use reasoning and truths already proven to not only determine if if a statement is true, but to prove it with an airtight argument. Alternatively, you can show something is false by identifying a counterexample. Success is a proof or counterexample; failure is a suspicion that you can’t prove either way.

I was good at the proofs, too.

I wanted to train for the William Lowell Putnam exam. The Putnam exam is a nationwide, all-day exam that poses 12 problems. The best possible score on the Putnam exam was 120 — full credit for each of the 12 problems. The median score is 1 out of 120. The problems are that hard.

To train, I took an advanced problems class in which we received three problems per week. There were four students in the class. The whole class consisted of the students presenting their solutions to the three problems. The professor would present only if the students were unable to solve one of the problems.

These were hard problems, like the Putnam problems. They were often simple to describe, but maddening to prove (or disprove). You had to use creativity, guile, and a wide variety of mathematical techniques. I attacked them with every tool I had: converging series, reductio ad absurdum (assume the opposite and prove it leads to a contradiction), mathematical induction, and concepts from every branch of math in which I’d taken classes.

My goal was to prevent the professor from ever having to present an answer. A typical student in the class would solve one or two of the problems in a week. My goal was to solve all three. And even if took all my spare time for the whole week, I set out to do that.

And I succeeded. I felt invincible.

The next semester, I took the class again. But the original professor had retired. A new professor took over the class. I was the only student. We met in his office. And once again, I set out to master all three problems every week.

About halfway into the class, I met my match.

The professor had posed an equation about ordinary numbers that looked entirely routine. My job was to prove or disprove that the equation held true in all cases.

All week, I worked with the equation. I tried every approach I could think of. It was completely clear that the equation was true; I could find no evidence of any situation in which it was false. And yet, no matter what I tried, there was no angle that would yield a proof.

Finally, after struggling all week, I arrived, sleepless and red-eyed, in the professor’s office. “You have stumped me,” I said. “I know this is true, but I cannot prove it. Please tell me the answer.”

The professor said “The proof uses complex variables.”

This made no sense to me. The equation was about ordinary real numbers. Complex numbers, like the square root of -1, didn’t enter into the equation. It was as if you had been asked to construct a doorway, and found out that the key tool you would need was a hand mixer. Why would you use complex variables to solve an equation like this?

“I have not yet taken the complex variables class,” I told the professor.

“I know,” he said. “I wanted to see if there was a proof that did not use complex variables. No one has ever found one. If there was one, I figured you would find it.”

At that point I became very upset.

This was not fair. The result of his deception was that my streak was broken. I could no longer say I had solved all the problems in the advanced class.

This felt wrong.

First off, the professor knew the answer, but presented the problem in a way that was misleading, since it required a technique that had nothing to do with the original problem.

Second, he exploited the fact that I did not have the training to solve the problem for his own benefit — to try to unearth an unpublished proof.

And finally, he asked me to find something no one had found before. Everything I’d done before was on paths that others had made. He’d left me in the wilderness and told me to hack my way out.

I’m sure I seemed very agitated, but somehow, I didn’t blow up. You didn’t do that with a professor, even if you were just sitting in his office. I took the three problems for the following week and left, dejected.

I’m not sure why this incident has stuck with me so much that now, 40 years later, I want to tell the story. I think it is because, that day, I learned something a lot more important than math.

I think a lot of us go through life with the attitude I had as an undergraduate. We know the tools we have, and the problems we need to solve. We know that a solution exists. And we need to find it.

But that is not how the world works.

Sometimes the universe, or your boss, or your government withholds a key piece of information.

Sometimes the only path is to go into the wilderness rather than following the paths trod before.

Sometimes you work for weeks, only to find that there is no solution.

This feels unfair. Just like me on that day with the professor, you feel like you’ve been handed a raw deal, deceived, messed with.

But when the problem is how to get your marriage to work, or how to deal with your addiction or your child’s depression or a medical diagnosis or to fix global warming, there’s not only no set of rules, there is no guarantee that if you do the right thing and work hard, you will succeed. The self-help books are wrong. Confidence is not all you need. Some things are just impossible.

Not only that, but there’s no way to know if they are impossible or not when you set to work on them.

I remain a very confident person. I continue to succeed at most things I set my mind to. But thanks to that class 40 years ago, I know that the “rules” may not be rules, the tools I need may be completely unexpected, and the problem may not, in fact, be solvable.

The work is still worthwhile. The solutions are satisfying. But the work, in the end, is what makes life worth living, regardless of whether a solution exists — or whether life is fair in the end.

15 responses to “Unfair

  1. Thanks for an interesting story. Perhaps the best spin on the 2nd professor’s behavior might be that he had been told of your extraordinary abilities by the first prof, and so the challenge to you was in one sense a compliment?
    …I know that interpretation is a stretch, but still it is possible….. Final consolation prize is your talent in math. I am quite jealous. Dare I say it is not fair that I don’t have this talent? 😉
    PS Thanks for your blog, it is terrific.

  2. Think about it this way: if you “knew” in advance that you weren’t supposed to be able to solve it, would you have tried as hard? I would bet that the effort you made benefited you in other ways.
    It also reminds me of the George Dantzig story–the grad student who solved a previously unsolved problem that the professor had written on the board.

    1. Love your post. It was an extraordinary compliment to you that turned out to be an invaluable life lesson. Thanks for sharing. Couldn’t have come at a better moment for me personally. And I loved the movie “Good Will Hunting”.

  3. Thank you for sharing this – it surely helps explain the precision behind your writing voice. I think we’d all like to hear how your Math Brain helped as you have gone through your career and how it shapes your writing today. Thanks again for writing this – ET

  4. For just $99/month, I can forward you a regular supply of SolutionSource aromatherapy oil that has been CLINICALLY DEMONSTRATED to activate the problem-solving faculties of the left Lowell cerebellum region, and to support the growth of mathematical variable resolution through combating free-radical Putnam variables.

  5. Sometimes not knowing something is believed to be insurmountable is what enables us to find a solution. It reminds me of a line from a Bob Bossin protest song, “But if we could end Apartheid and save Clayquot Sound, we can ….” We are capable of more than we know.

  6. Ahh, this is my approach with summer interns working at our software company. Interns get two kinds of assignments: the ones that are tedious and no one WANTS to do, and for the interns that appear to be super-capable, the ones that we think are too hard. The thing about the second type is that we don’t tell the interns that we think they are hard; we are really just seeing if they have another approach that our world-weary brains don’t consider. And more than half the time (albeit with a very small sample size), it works!

    I agree with the others, this was a poorly-expressed compliment from your professor

  7. Fantastic article. I too started out as a Math major in college. The junior and senior classes were actually fun to me!

    But I agree when you’re an overachiever you think if I do X then I will get the result Y.

    Most recently my mom was in the hospital with a bad infection. She wasn’t able to talk. I wanted to find X to get the result of her continuing to live and come home. The doctors clearly were far better trained in finding X than I was.

    It was frustrating to be so helpless. I take care of my mom since she lived with me. As long as I fed her and had her caretakers give her baths, clean her room and do her laundy she should keep living happily.

    In this case I guess God has the complex variables and that’s the binary simplicity that he did not engineer hearts or bodies to last forever.

    I watched the numbers on the heart rate monitor decrease. 110, 92, 80, 75, 62, 54, 45…. 0. Then bumped up to 38. Then down to 0 again. And they shut off the machine and declared the time of death.

    And despite God creating the natural world with certain evident mathematical sequences. Time does eventually run to 0.

  8. I have a different take on this. I think it was a power play by the professor and you were justifiably angry. I imagine you no longer prided yourself on getting all three problems done each week, in case they were also beyond you and unsolvable. So for the sake of his ego he took you down a peg, and likely diminished your enthusiasm for math.

  9. We have a saying in tech hiring. “Hire smart people young while they still don’t know what they are not capable of accomplishing.”

    I actually read what he did to you as a huge compliment. That as far as all the mathematicians in the world go they couldn’t find a way to prove it without complex variables.

    And if there were anyone in the world who could you are The One. The Golden Child. He’s saying you are better at raw math than he himself was.

  10. I am fascinated by the suggestions from all of you about the motives from the professor.

    I don’t think he wanted to take me down a peg. I was one of the best math students there in years, I am certain that he and the other profs wanted to encourage me. With their help, I went on to become a Ph.D. student at MIT.

    I do think that he wanted to see if there was a proof that used real numbers. Unlike the other prof, he was willing to test my skills on a problem for which there was not a known solution. I think he wanted to introduce me to the real world of mathematical research, which is where I was headed, and where you don’t know if a solution even exists.

    And in doing that, he was successful. Until I hit a real unsolvable problem in grad school and got off the academic track, that is.

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