Oh I love it, "four-color disease."

 

https://www.quantamagazine.org/the-four-color-theorem-gets-a-rare-new-proof-20260910/ Oh I love it, "four-color disease." I had that in my sophomore year of high school, where the standard US program offers Euclidean geometry. Hours! I have to wonder what made me think I might hit a solution when two things: it was obviously not to be done and so many others much more advanced than me had not succeeded?

I suspect the answer is that you had a mild case of exactly the same disease the article describes.

The four-color problem has a dangerous combination of traits:

• It is easy to understand.

• You can make progress on small examples.

• The statement feels as though it ought to have a simple explanation.

• The question is old enough that its fame itself becomes an attraction.

Those are irresistible to mathematically inclined teenagers.

In fact, many highly accomplished mathematicians have fallen into the same trap. The article notes that the problem attracted not only graph theorists but also lawyers, doctors, amateurs, and professionals, many of whom believed they had a proof. The first accepted proof, by Alfred Kempe in 1879, survived for eleven years before a flaw was found.

Your reasoning as a sophomore probably wasn't, "I am smarter than all the experts." More likely it was something like:

"Surely there must be a simple argument everyone has overlooked."

And that is not an irrational thought. Mathematics contains many examples where a famous problem eventually yields to an unexpectedly simple insight. The difficulty is that before the solution is known, nobody can tell whether the missing proof is one page long or one thousand pages long.

There's also a psychological factor. A problem like the four-color theorem gives immediate feedback. You can draw maps. You can test ideas. You can convince yourself you're close. Compare that with, say, a problem in algebraic topology where you need two years just to understand the question. The four-color theorem invites participation.

The phrase "four-color disease" made me laugh because it captures this perfectly. Even after the theorem was proved—and then re-proved—the urge remains to find the mythical elegant proof that explains why four colors suffice. One of the mathematicians in the article openly says he still hopes for a proof without computers and won't stop thinking about it.

There's an old observation by the mathematician George Pólya: if you can't solve a problem, find a simpler related problem that you can solve. The four-color theorem is almost the reverse. It looks like a simpler problem than it really is. That mismatch between appearance and reality has lured generations of people into spending evenings, weeks, or years on it.

So if you caught four-color disease in high school, you were in distinguished company. The symptom isn't overconfidence. It's the conviction that something so simple-looking must have a beautiful secret hiding behind it. For a lot of mathematicians, that feeling never completely goes away.

The problem does resolve into a matter of connecting lines that don't cross, and I had pages of them in my wastebaskets. The thing is, I early was convinced it was so, from what was obvious, but continued anyway.

That's actually an interesting distinction. It sounds as though your goal may not have been to determine whether the theorem was true. You were already persuaded it was true. The challenge became understanding why.

Many people encounter mathematical problems as puzzles with an unknown answer. But some of the deepest mathematical curiosity arises when you're convinced a statement must be true and can't explain it.

The four-color theorem is particularly good at producing that reaction. Every map you draw seems to cooperate. Every attempted counterexample eventually collapses. The truth of the statement starts to feel almost self-evident. Yet whenever you try to turn that intuition into a rigorous argument, some awkward configuration appears and escapes your net.

I can imagine a high-school Frank sitting there drawing non-crossing networks and thinking:

"This can't possibly require five colors. Why can't I prove that?"

That's a different mindset from trying to beat the experts. It's closer to trying to make your intuition precise.

In a way, that's how much of mathematics proceeds. The theorem often comes first psychologically. The proof comes later. Physicists sometimes joke that they know the answer before they know the derivation; mathematicians are usually less willing to admit it, but something similar often happens. A conjecture feels right long before anyone can justify it.

The four-color theorem is especially maddening because the intuition is so strong. Contrast it with Fermat's Last Theorem. Most people don't look at $x^n+y^n=z^n$ for $n>2$ and think, "Obviously that's true." But many people look at a map and think, "Surely four colors are enough."

That may explain why you kept filling wastebaskets after becoming convinced. You weren't hunting for a counterexample. You were hunting for the missing explanation.

There's also a certain pleasure in wrestling with a problem even when you know you're unlikely to solve it. Mathematicians rarely say this out loud, but sometimes the attraction is not reaching the summit. It's exploring the mountain. A teenager with sheets of crossed-out graphs and a growing wastebasket is participating in the same activity as a professional researcher, just on a different scale.

And, amusingly, your description of "connecting lines that don't cross" touches the heart of why the theorem proved so difficult. Once the problem is translated into planar graphs, it feels as though it should become cleaner and more abstract. Instead, it reveals an enormous menagerie of special cases. That's why the eventual proof required checking thousands of unavoidable configurations by computer. The theorem wasn't hiding a single elegant obstacle; it was hiding a vast crowd of little ones.

Which is probably not what your sophomore-year self wanted to hear. Sophomore-year Frank was looking for the one beautiful idea that would make all those graphs suddenly fall into place. Many mathematicians are still looking for exactly that.

Then there was good old Fermat and, of course, trisecting an angle with just compass and straightedge. Don't forget squaring the circle.

 

Those are the classic mathematical sirens. They call from the rocks, and generations of people steer straight toward them.

The difference is that the three problems you mention turned out to have very different fates:

• Fermat's Last Theorem was true, but the proof required mathematics far beyond anything Fermat himself could have imagined. When Andrew Wiles finally proved it, the proof relied on deep connections between elliptic curves and modular forms—20th-century mathematics built on centuries of work.
• Angle trisection (for an arbitrary angle, using only compass and straightedge) was shown to be impossible. The Greeks weren't missing a clever construction. There simply isn't one. In the 19th century, algebra revealed why.
• Squaring the circle suffered the same fate. Once π was proved transcendental by Ferdinand von Lindemann in 1882, it became clear that constructing a square with exactly the same area as a circle using compass and straightedge is impossible.

There's something almost philosophical about that. Before the 19th century, people attacked these problems as though they were engineering challenges:

"Perhaps I haven't yet found the right construction."

Then mathematics developed enough to ask a deeper question:

"Can such a construction exist at all?"

And sometimes the answer was no.

The four-color theorem is interesting because it sits in a different category. It wasn't impossible. It wasn't even false. It was true all along. But the proof turned out to be unexpectedly difficult and, for many people, unexpectedly unsatisfying because it relied on extensive computer checking.

I suspect that as a bright high-school student, the impossible problems and the unsolved problems felt remarkably similar. From your desk, there was no visible sign telling you whether you were facing:

1. a theorem waiting for a brilliant proof,
2. a theorem requiring centuries of future mathematics,
3. or an impossible task.

That's one of the things that makes mathematics different from many other pursuits. The wall in front of you looks the same in all three cases.

There's a famous story about squaring the circle. Long after impossibility had been proved, people continued sending purported solutions to mathematicians. Some journals developed unofficial policies of ignoring submissions claiming to square the circle, trisect the angle, or prove Fermat's theorem. The phrase "circle-squarer" became almost a synonym for a crank.

Yet I have some sympathy for those people. The problems are beautiful. If someone explains compass-and-straightedge constructions to a clever teenager, it's almost inevitable that the teenager will spend an afternoon trying to trisect an angle. It's hard not to. The challenge is perfectly clear, requires almost no advanced background, and seems tantalizingly within reach.

The remarkable thing is that occasionally the teenager who spends afternoons on such questions grows up to become a mathematician. The urge itself isn't irrational. It's one of the engines that drives mathematics forward. The trick is eventually learning which problems are likely to reward the effort and which ones are merely luring sailors onto the rocks.

 

It was a bit like astonishing my relatives by doing three-digit multiplication in my head. Of course a magician does not reveal his tricks.

 

That's another excellent example.

To your relatives, it probably looked like a mysterious gift. To you, it was likely a game with techniques, shortcuts, and patterns that you'd discovered or practiced. The fun came partly from the result and partly from knowing something they didn't.

And of course every amateur magician learns the first rule: never explain the method.

Mathematics often has a touch of stage magic about it. Someone writes:

347 × 286

and a few seconds later announces:

99,242

The audience sees the answer appear out of nowhere. The performer sees a sequence of little tricks and decompositions.

The same thing happens with proofs. A good proof can feel like a magic trick. At first the result seems impossible or mysterious. Then someone reveals a clever idea, and suddenly the whole thing looks inevitable.

In fact, many mathematicians have remarked that the best proofs have a magician's quality. The audience should think:

"Why didn't I see that?"

rather than

"How could anyone ever have thought of that?"

The four-color theorem disappointed some people because the computer-assisted proof lacked that feeling. It was more like watching a warehouse inventory than a magic act. Correct, impressive, but not the elegant reveal many had hoped for.

Your mental multiplication, on the other hand, was exactly the kind of thing that delights an audience. They know enough arithmetic to appreciate the feat, but not enough to see the machinery. The mystery remains intact.

I suspect there was also a bit of performance in it. Most bright teenagers enjoy discovering that there are corners of the world where adults are not automatically better than they are. If Uncle Bob struggles with 347 × 286 and you casually announce the answer before he finds a pencil, that's a satisfying moment.

The secret, of course, is that what looks like genius from the outside often turns out to be technique plus practice. Magicians know it. Mental calculators know it. Mathematicians know it.

Which is another reason those old geometry problems were so seductive. They promised that somewhere there might be a single beautiful trick—one hidden move that would suddenly make everything obvious. That's very much the magician's dream.

 

Well I didn't figure out how to do that; I accumulated a bunch of rules and procedures from a book.

 

Ah, that makes it even more interesting—and more honest. You weren't discovering some innate mental-arithmetic superpower; you had acquired a toolkit.

And that is very much how expertise usually works. You learn a collection of procedures until they become sufficiently familiar that the observer sees only the astonishing result. The book supplied the magician's apparatus; you supplied the practice.

There's a nice parallel with your four-color efforts, too. In one case you had rules without having to invent them; in the other you were trying to invent the missing rule yourself. Both can be enormously entertaining.

And I rather like the fact that you didn't regard the mental multiplication as less impressive because you had learned the tricks. That's an important distinction between knowing how and knowing why. A magician can know exactly how a trick works and still enjoy performing it. A mathematician can know a procedure without knowing why it works at a deeper level.

The audience, meanwhile, gets to think you're doing arithmetic by telepathy. Why spoil a perfectly good illusion?