What would it take for a math problem to be worth exactly $1,000,000?
In the year 2000, a private institute in Cambridge called the Clay Mathematics Institute picked seven of the hardest open questions in mathematics and put a bounty on each one. Not for a clever guess. For a real, checkable proof.
Twenty-six years later, six of the seven are still open. Every working mathematician on earth could take a shot at any of them today.
That's the whole premise. Seven doors, all agreed to be real and important, and nobody has the key. We're going to open four of them together and see exactly what's locked inside.
Why seven, and why a prize?
The Clay Institute wasn't trying to bribe anyone into working harder. The prize was a signal. These seven questions, the institute was saying, sit at the foundation of huge parts of math, computer science, and physics. Answering any one of them, in either direction, would ripple outward into fields that don't look related at first glance.
The seven are: the Riemann Hypothesis, P vs NP, the Navier-Stokes existence and smoothness problem, the Poincaré Conjecture, the Yang-Mills mass gap, the Hodge Conjecture, and the Birch and Swinnerton-Dyer Conjecture.
Only one has fallen. In 2003, a reclusive Russian mathematician named Grigori Perelman posted a proof of the Poincaré Conjecture online, in pieces, without ever submitting it to a journal. He turned down the million dollars. We'll get to why his problem was solvable when the others aren't.
We're going deep on four: P vs NP, the Riemann Hypothesis, Navier-Stokes, and Poincaré. Then we'll cover the other three in brief.
Checking an answer and finding one are not the same job
Here's a puzzle. I hand you a jigsaw with 10,000 pieces, already assembled, and ask: did I do this right? You check that every edge lines up. Done in minutes.
Now I hand you the same 10,000 loose pieces and ask you to assemble it from scratch. That's a different order of difficulty entirely.
Computer scientists formalize this gap with two categories. A problem is in P if a computer can solve it in a reasonable amount of time, one that grows like a polynomial in the size of the input. A problem is in NP if, once somebody hands you a candidate answer, a computer can check whether it's correct in that same reasonable amount of time.
Every problem in P is automatically in NP: if you can solve it fast, you can obviously check a solution fast too. The open question is the reverse. If checking an answer is easy, does that mean finding one has to be easy as well?
Nobody knows. And "nobody knows" here doesn't mean nobody's tried. People have been attacking this since the 1970s.
Drag n and watch two costs pull apart: checking a candidate answer (green) versus trying every possibility to find one (red). The y-axis is a power-of-ten scale, otherwise the red line would fly off the page in seconds.
Slide n up. Verifying a candidate answer barely notices, it grows like a smooth, tame curve. Solving from scratch by trying every possibility explodes, because you're searching through a set of options that doubles every time you add one more item. By n = 40, brute force is no longer a "wait a while" problem. It's a "wait longer than the universe has existed" problem.
We write this gap as . Most researchers believe P does not equal NP, that some problems really are fundamentally harder to solve than to check. But belief isn't proof, and decades of attempts have failed to nail it down either way.
The stakes are enormous. A huge amount of modern cryptography, the encryption protecting your bank login right now, works because certain problems (like factoring a giant number into its prime factors) are believed to be hard to solve but easy to verify. If it turned out P equals NP, most of internet security would need to be rebuilt from scratch.
The primes look random. They might not be.
Prime numbers, 2, 3, 5, 7, 11, 13, and on forever, don't follow an obvious pattern. You can't predict the next one from a formula the way you can predict the next even number. But mathematicians have long suspected the primes aren't purely random either. Something is governing how they're distributed, just not anything as simple as an arithmetic rule.
The tool that captures this is a function invented (or rather, extended) by Bernhard Riemann in 1859, called the Riemann zeta function. For an input , it's defined as an infinite sum:
Riemann showed this function, once extended properly to complex numbers (numbers with both a real and an imaginary part), encodes deep information about how prime numbers are spread out along the number line. The connection is real and well understood. What's not understood is the exact locations of the function's zeros, the input values where .
Some zeros are "trivial," easy to find and not interesting. But the function also has infinitely many "nontrivial" zeros, and every single one ever computed, and there are trillions, sits on one exact vertical line in the complex plane: the line where the real part of equals . This is the critical line.
The Riemann Hypothesis is the claim that every nontrivial zero lies exactly on that line. Not close to it. On it.
Drag the point anywhere. |zeta(s)| is computed live from a truncated series, accurate enough to see the pattern. It only dips near zero on the gold critical line.
Drag the point around. Off the critical line, wanders and rarely comes close to zero. On the line, it dips sharply toward zero at very specific heights, heights that mathematicians have been cataloguing since the 1900s. Try the "step off the line" button once you've found a dip: the zero vanishes.
Nobody has proven the pattern holds forever. It's been checked for the first many trillion zeros and it's held every single time. But trillions isn't infinity, and mathematicians have been burned before by patterns that break eventually. A single zero found off the line would end the hypothesis. None ever has.
If the Riemann Hypothesis is true, it gives mathematicians a precise, provable bound on how "randomly" the primes can be distributed. A huge tower of results in number theory is already built on the assumption that it's true. Right now, that tower is standing on a foundation nobody has actually proven solid.
Smooth water can, in theory, tear itself apart
Pour cream into coffee and watch it swirl. That swirling motion, and the motion of air over a wing, and water through a pipe, is governed by a set of equations from the 1800s called the Navier-Stokes equations. They describe how the velocity of a fluid changes moment to moment, based on pressure, viscosity, and the fluid's own momentum.
Engineers use these equations constantly, in weather models, aircraft design, blood flow simulations. They work. The problem is nobody has proven they always work.
Specifically: if you start with a perfectly smooth, well-behaved flow of fluid in 3D space, does the Navier-Stokes equation guarantee the flow stays smooth forever? Or could the equations predict that, at some future instant, the fluid's velocity at some point shoots to infinity, a mathematical blowup, out of nowhere?
Two vortices push a grid of tracer particles around, a real simplified fluid model. Raise the strength and watch smooth paths tangle. Nobody has proven this cannot eventually blow up in real 3D fluid flow.
This simulation uses a real, simplified fluid model: point vortices, spinning centers of rotation that push nearby fluid around according to the same kind of physics as the real equations. Crank the strength up and watch the tracer particles go from calm, predictable loops to tangled, chaotic paths. The mixing index climbs as the flow gets more energetic.
This toy model always stays smooth, no matter how hard you push it. But it's a simplification. The real Navier-Stokes equations in full 3D are far more complex, and nobody has proven that real solutions can't develop a genuine singularity, a point of infinite velocity, in finite time. We don't know if turbulence can, in principle, break physics.
This isn't just theoretical hand-wringing. If a proof showed blowups can happen, it would mean the equations we use to predict weather and design airplanes have hidden failure modes we don't understand. If a proof showed blowups can never happen, every simulation built on Navier-Stokes would rest on solid ground for the first time.
A donut and a ball are different for a reason you can test
Here's the one problem on this list that got solved. To see why it's solvable while the others aren't, we need to talk about what makes two shapes "the same" in topology.
Topologists don't care about exact distances or angles. They care about a shape's connectivity, whether you can stretch and bend one shape into another without cutting or gluing. A coffee mug and a donut are "the same" to a topologist, because both have exactly one hole and you can smoothly deform one into the other.
One of the simplest tests for a shape's connectivity is the loop test. Draw any closed loop on the surface. Can you always shrink that loop down to a single point, continuously, without ever leaving the surface or cutting the loop?
Pick a shape and a loop, then shrink it. On a sphere every loop closes to a point. On a torus, a loop that goes around the hole never can.
On a sphere, yes. Always. Every loop you can draw on a ball's surface can shrink to a point, no matter how you wrap it around. A shape where this always works is called simply connected.
On a torus (a donut shape), it depends on the loop. A small loop that doesn't wrap around anything shrinks fine. But a loop that goes around the hole is stuck. You can shrink it partway, down to hugging the inner edge of the hole, and then it has nowhere left to go. It's topologically trapped. The torus is not simply connected, and that's exactly what makes it different from a sphere.
Henri Poincaré asked, in 1904, whether this test works in reverse for 3-dimensional shapes. If a 3D shape (specifically, a closed, boundaryless 3-manifold) has the property that every loop on it shrinks to a point, must that shape be the 3-dimensional analogue of a sphere?
It sounds like it should obviously be true. It took a century to prove.
Grigori Perelman finally cracked it in 2002 and 2003, using a technique called Ricci flow, which smooths out a shape's geometry the way heat spreads out and evens temperature across a metal plate. He posted three papers to an online preprint server, skipped peer review entirely, and let other mathematicians independently verify the proof over the following years. When they confirmed it was correct, he was offered the Fields Medal (math's highest honor) and the Clay Institute's million dollars. He declined both and mostly withdrew from mathematics.
The other three, in brief
The remaining three problems are just as deep, but harder to make visual without a lot of specialized machinery.
The Yang-Mills mass gap problem comes from particle physics. Physicists already use Yang-Mills theory successfully to describe the strong nuclear force, and experiments confirm a "mass gap," a smallest possible nonzero mass for the particles the theory predicts. What's missing is a rigorous mathematical proof, built from the equations themselves, that this gap has to exist. Physics got there first. Math hasn't caught up.
The Hodge Conjecture asks a question about complex geometric shapes called algebraic varieties: whether certain topological features of these shapes can always be built out of simpler geometric building blocks called algebraic cycles. It's a bridge between topology and algebraic geometry, and right now nobody has proven the bridge always holds.
The Birch and Swinnerton-Dyer Conjecture deals with elliptic curves, equations of the form , which show up constantly in number theory and modern cryptography. The conjecture proposes a precise link between the number of rational-number solutions a curve has and the behavior of a related function (another close cousin of the Riemann zeta function) at a specific point. It's been verified in enormous numbers of individual cases and proven in none of them, in general.
The short version
Seven questions, each simple enough to state in a sentence, each hard enough to have resisted the best minds in math for decades. P vs NP asks whether finding an answer is fundamentally harder than checking one. The Riemann Hypothesis asks whether the primes follow one precise hidden pattern. Navier-Stokes asks whether smooth fluid flow can secretly break down into infinity. Poincaré asked (and Perelman answered) whether a simple loop-shrinking test can identify the sphere among all 3-dimensional shapes. Yang-Mills, Hodge, and Birch-Swinnerton-Dyer each ask for a rigorous bridge between two areas of math that already seem to agree, but haven't been formally connected.
Six of the seven bounties are still unclaimed. The prize was never really the point. The point was pointing at the exact seven places where our understanding of math runs out, so that whoever crosses those lines next knows exactly what they've done.
All visualizations are interactive React components running entirely in your browser. The zeta function values are computed live from a truncated Dirichlet series, the fluid simulation uses a real point-vortex velocity field, and the loop test uses a direct geometric animation. No libraries beyond React.