When a result is not a result... or is it?

I had a big scare last week.

In late September, I managed to solve a problem I'd wanted to solve for a long time. Since then, I've managed a few related results whose upshot is that my solution is the best that can be found, in the context I'm considering—for the logically literate, that means that this result is "sufficient and necessary"—and I'm assembling them into a paper that I hope to submit Real Soon Now^TM.

On Wednesday, I was checking the paper, and I got stuck on one of the details; I had written, t_m where I should have written t_m - 1. From such small errors are great panics made.

After a few hours' combination of staring at it and trying to convince myself that what I had written, checked at least twice, and typed (checking again, presumably) really was correct, I had to conclude that it was wrong.

A dark sense of dismay sank into my heart, filling it with a cold fog. Wednesday looked really bad by the time I walked out of the office to give my last Calculus lecture of the semester. Several months' worth of work looked completely useless, all because of a careless error that I shouldn't have made in the first place, let alone after checking, re-checking, and thrice-checking while typing. I had to calculate the gcd (greatest common divisor), and instead I calculated the lcm (least common multiple). Pfah. Even grade schoolers know the difference.

It did occur to me that there might be a light at the end of the tunnel. Finding a mistake meant that the true solution was still out there, waiting to be found, and that by making the appropriate change, I could find it. That would give me an even better result than the first one, since the second one would have the virtue of being correct! Of course, that's not very reassuring when you've spent more than two months thinking your previous solution was correct.

In any case, it would have to wait until after Calculus. I hate waiting when a mistake clouds the mind's vision. I hate being wrong.

Some students followed me back to my office. My obsession with the problem had so clouded my thoughts that I had forgotten to bring a recently graded test to class. I handed their tests to them and answered a few questions.

Alone again, I began from the premise that yes, I had made a mistake, and I needed to discover the correct solution. It looked impossible at first, and I wondered if the entire approach was wrong. Since this approach was the only one that made any sense at all, I began to feel worse.

Then I noticed something. It was unlikely and surely impossible. But, after checking it—a little more carefully this time, I hope— I concluded that it was not merely possible, but true, splendidly true, even. I even proved it a second way just to be sure. In addition, it corresponds neatly with the previous proof of the necessity—something the previous ("wrong") result hadn't done. I'd never really noticed that discrepancy until I found the mistake in the proof of the sufficiency.

By the time I left, I had a result again. A bad Wednesday, but it could have been worse. Needless to say, I will be check, re-check, and extra-check the result again.