The Feynman Trick That Stumped a Math Competition

I cover biotech and genomics. I also cannot stop thinking about this integral. Bear with me.

I'm a biotech reporter. I spend most of my days thinking about CRISPR and RNA folding and whether any given startup is going to survive its Series B. And yet here I am, at an embarrassing hour, re-watching footage of two people trying to solve a calculus problem under competition pressure, completely unable to look away.

The integral in question comes from the Heidelberg Integration Bee, a recent competition where physicist and YouTuber Richard Behiel submitted a puzzle of his own design. Behiel posted a walkthrough of it — hint, competition footage, full solution — and I need to tell you about it because it does something I didn't expect: it's not really just about math.

It's about what happens when knowledge is a locked door. And who gets handed the key beforehand.

The Trap Inside the Problem

Let me give you the setup without making you cry.

The integral looks ferociously complicated. The denominator is this long, unwieldy series: x minus x²/2 plus x³/3, and so on. It looks like something designed to make you panic.

It isn't, though. Behiel explains: "The denominator of the integrand is the Taylor expansion of natural log of x. All it is is the recognition that the denominator is a fancy way of writing negative natural log of x."

That's step one. You have to see that this impossible-looking thing is actually a disguised version of something simple. If you've memorized that particular Taylor series expansion — if it lives in your pattern-recognition library — the door swings open immediately.

If it doesn't? Behiel is cheerful about it: "If they don't, well, I guess then they're stuck."

Yeah.

The problem was deliberately dressed up this way. Behiel says he originally submitted a cleaner version; the competition wanted it to "look a little more zesty." That's a fascinating design decision. The added complexity isn't mathematical depth — it's camouflage. It's a test of whether you've seen this specific pattern before.

And I can't just let that sit there without saying something: that's not a test of mathematical intelligence. It's a test of mathematical exposure.

Those are related. But they are not the same thing.

Who gets drilled on Taylor series expansions until they're reflexive? Students with rigorous prep, access to good teachers, problem sets from competitive math training programs. The gate on this integral — Behiel's word, and it's the right one — filters for background as much as brilliance. That matters. Not because competition math should be easy, but because we should be clear-eyed about what we're actually measuring when we watch people compete under these conditions.

Step Two: The Part That Made Me Lose It

Okay. Once you've cleared the Taylor series gate, you're facing a different monster.

The integral now involves x raised to the power e². Not e raised to something. x raised to e². Behiel captures the dissonance perfectly: "Normally, you think about e to the x, e to the x², e to the x to the -2, but x to the e to the something, what? That's backwards. That's wrong. How are you supposed to deal with that?"

This is where the Feynman trick comes in. 🧬 (Yes I'm using that emoji in a math piece. The elegance here is genuinely biological-feeling, okay?)

The technique works like this. Instead of trying to solve the integral directly, you define a new function — call it I(t) — where you've replaced the constant e² with a variable t. The original integral is just I(e²). Now, rather than solving for I directly, you differentiate it with respect to t.

And something gorgeous happens. The logarithm that was making your life miserable cancels out — it appears in both the numerator and denominator and just... dissolves. You're left with the integral of xᵗ dx, which is the most reasonable thing in the world. That evaluates to 1/(t+1).

So now you know I'(t) = 1/(t+1). You integrate that. You evaluate it between e⁻² and e² (choosing e⁻² as your lower bound because I(e⁻²) = 0, which Behiel shows by inspection — the two numerator terms cancel). You get ln((e²+1)/(e⁻²+1)).

Then — and THIS is the part — you factor e² out of the numerator and realize you're looking at e²·(1+e⁻²) over (e⁻²+1).

Those are the same thing.

They cancel.

You're left with ln(e²).

Which is 2.

The answer is 2. This sprawling, intimidating, disguised monster of an integral. The answer is just 2. I actually said "oh come ON" out loud. Your brain spends all this effort climbing what looks like a cliff face and then you're standing on completely flat ground going — wait, that's where I ended up?

That's the Feynman trick doing what it does. You make the problem temporarily more complicated (introduce a variable, take a derivative) so you can make it permanently simpler on the other side. It's the calculus equivalent of taking a longer route to avoid traffic and realizing the longer route is actually faster.

What the Competition Footage Actually Reveals

Behiel includes footage of two contestants, Loots and Juan, working through the problem under time pressure.

Loots starts by trying to expand the series in sigma notation and work through a u-substitution. Juan gets further — he spots the Taylor series disguise early, which is the critical first insight. But he drops a minus sign. Then he erases his denominator (Behiel's commentary: "No, don't erase the denominator"). Then he tries his own u-substitution, defining u = xe². It's a reasonable instinct. It doesn't work. Time's up.

Behiel is genuinely generous about this, and I think he's right to be:

"If you put me up there, I got five minutes, everyone's watching, I'm not going to necessarily be able to recognize a Taylor series of negative log x or realize that you got to do a Feynman trick. Like I feel like when you're on the spot with all that attention, all the time constraints, like integrals become way harder."

This is worth taking seriously. The competition format — time-pressured, high-visibility, filmed — is its own kind of variable in the problem. Juan clearly knew something was happening with that xe² term. Under different conditions, with more time and no camera, maybe he gets there.

Behiel also notes that Loots did identify the Taylor series correctly (mostly — close enough). Both contestants showed real mathematical instincts. Neither showed the specific combination of pattern-recognition and trick-retrieval the problem required, under those conditions, in that window.

Which brings me back to the gate.

The Feynman trick is genuinely beautiful. It's the kind of mathematical move that, once you've seen it, you want to apply it to everything — you start looking at integrals and wondering what happens if I just introduce a parameter here and differentiate. That itch is real and it's good.

But "once you've seen it" is doing a lot of work in that sentence.

Behiel's video is a fantastic piece of math communication — clear, honest about competition pressure, genuinely exciting to follow. And the puzzle he designed is a good one. But watching it, I kept thinking: the students who cruise through this first gate aren't necessarily smarter. They're more prepared. Often because they had more access to preparation.

Competitive math has a pipeline problem that looks like a talent problem from the outside.

The answer being 2 is almost taunting in its simplicity. All that complexity, perfectly engineered to dissolve. The question of who gets taught how to dissolve it — that one doesn't resolve so cleanly.

Mei Zhang writes about biotechnology, genetics, and the occasional calculus competition that won't leave her brain. She covers the future of medicine — and apparently everything that touches the question of who gets access to knowledge.