Seven problems.
Seven million dollars.

The everyday version

Think about a hard Sudoku. Filling it in can take you an hour of trial and error. But if a friend hands you a completed grid, you can tell in under a minute whether it is right. Solving is hard; checking is easy.

That gap shows up everywhere. Fitting a delivery round into eight hours, seating 200 wedding guests so no feuding relatives share a table, packing a van, folding a protein, scheduling exams — in every case, a proposed answer can be verified almost instantly, and yet finding one seems to require sifting through an astronomical number of possibilities.

P vs NP asks whether that gap is real. Is searching genuinely harder than checking, or have we simply never found the clever shortcut?

What is actually being asked

P is the class of problems a computer can solve quickly — in polynomial time. NP is the class where a proposed solution can be checked quickly. Every problem in P is obviously in NP. The question is whether the reverse holds:

Does P = NP?

There is a twist that makes the question sharper. In 1971 Cook showed that certain problems in NP are NP-complete: universal, in the sense that a fast algorithm for any single one of them would immediately give a fast algorithm for all of them. Thousands of practical problems — from circuit design to Tetris — are now known to be NP-complete. They all stand or fall together.

Why anyone cares

Nearly all modern cryptography is a bet that P ≠ NP. Your bank's encryption works because multiplying two big primes is easy while pulling them apart again is not. If P = NP — and the algorithm were practical rather than merely polynomial — that bet collapses.

The upside would be just as vast. Optimal drug design, perfect logistics, chip layouts that cannot be improved, and mathematics itself partly automated: if verifying a proof is easy, and P = NP, then finding a proof of any theorem with a short proof becomes easy too.

Where it stands

Wide open, and most researchers expect the answer to be no. In a 2019 poll of experts, close to nine in ten said they believe P ≠ NP. What makes it so stubborn is that the field has proved several barrier theorems: entire families of proof technique have been shown, rigorously, to be incapable of settling the question. Any solution will need an idea nobody currently has.

Try it → Solve the puzzle yourself, then hit Brute force and watch a computer do it the dumb way. Then drag the size slider and see how fast “dumb” becomes impossible.

The everyday version

Primes — 2, 3, 5, 7, 11, 13 — are the atoms of arithmetic, and they arrive on the number line with no obvious pattern. But step back far enough and a pattern appears: primes thin out at a predictable rate. Near a large number x, roughly one in every ln(x) numbers is prime.

That is the smooth trend. The interesting question is the error: how far do the actual primes wander from it? Riemann found something extraordinary. The error is not noise. It is a sum of waves — and every wave is generated by one special number, a zero of the zeta function. Add up enough of these waves and the smooth curve grows steps, and the steps land exactly on the prime numbers.

Every zero anyone has ever computed sits on one vertical line in the complex plane. Riemann guessed they all do. That guess is the hypothesis, and it is equivalent to saying the primes are distributed as evenly as they possibly could be — the error never exceeds roughly √x.

What is actually being asked

The Riemann zeta function starts as an innocent sum and is then extended to every complex number:

ζ(s) = 1 + 1/2^s + 1/3^s + 1/4^s + …

It vanishes at the negative even integers −2, −4, −6 … (the trivial zeros, which nobody worries about). The conjecture concerns all the others:

Every non-trivial zero of ζ(s) has real part exactly ½.

Why anyone cares

Hundreds of published theorems begin “assume the Riemann Hypothesis”. An entire shadow literature exists of results that are true if it holds. Proving it would promote all of them to fact in a single stroke; disproving it would demolish them.

There is also a strange physical echo. The spacing of the zeros matches the statistics of energy levels in heavy atomic nuclei, described by random matrix theory. Nobody knows why number theory and quantum chaos should share a fingerprint.

Where it stands

Open since 1859, and Hilbert's eighth problem in 1900 as well. Hardy proved in 1914 that infinitely many zeros lie on the line; later work showed a positive proportion do. More than ten trillion zeros have been checked by computer, and every one obeys. That is powerful evidence and no proof at all — number theory has a history of patterns that hold for astronomically long and then fail.

Try it → Drag the slider from zero waves upward and watch the staircase of primes assemble itself out of the zeros. Then switch to the second tab to see the zeros themselves, found live by your browser.

The everyday version

Electromagnetism has unlimited reach. You can see stars because photons, the particles of light, are massless and so can carry the force across billions of light years.

The strong nuclear force is built from a theory of the same shape — a Yang–Mills theory — and its force carriers, the gluons, are also massless in the classical equations. By the same logic it should reach across the cosmos. It does not. It dies out after about a millionth of a billionth of a metre.

The accepted explanation is that the quantum version of the theory behaves nothing like the classical one. Gluons pull on each other, tangle up, and the lightest object the theory can actually produce — a glueball — has real, positive mass. A force carried by something heavy has a short range. That minimum mass is the mass gap.

Every experiment agrees. Every supercomputer simulation agrees. There is no proof, and worse: nobody has yet managed to write down the quantum theory itself in a mathematically rigorous way in four dimensions.

What is actually being asked

For any compact simple gauge group, prove that a quantum Yang–Mills theory exists on four-dimensional space–time and has a mass gap Δ > 0.

Two demands in one. Existence: construct the theory to the standards of rigorous mathematics, satisfying the standard axioms of quantum field theory. Mass gap: show the lightest excitation above the vacuum weighs something.

Why anyone cares

This is the missing foundation under a third of the Standard Model. Physicists calculate with quantum field theory constantly and to spectacular accuracy, yet the four-dimensional theories they use have never been shown to exist as mathematical objects. A proof would also explain confinement: why no one has ever seen a lone quark.

Where it stands

Open. Rigorous constructions exist in two and three space–time dimensions; four is the hard case and the physical one. Lattice simulations compute the glueball mass to a few per cent, so the answer is not in doubt — only the proof.

Try it → Click in the field to disturb it. With the mass at zero the ripples run away forever. Nudge the mass up and watch the disturbance refuse to travel — that is a short-range force, appearing in front of you.

The everyday version

The Navier–Stokes equations are Newton's laws written for fluids. They govern smoke, blood, ocean currents, the air over a wing, the weather. They are about two hundred years old and they work: every weather forecast you have ever seen came out of them.

Here is the embarrassment. Start a fluid off in a perfectly smooth state and let the equations run. Do you always get a smooth, sensible answer for all time? Or can the maths tie itself into a knot so tight that some quantity — the speed at a point, say — races to infinity in a finite amount of time?

That event is called a blow-up, or a singularity. Nobody has ever produced one. Nobody has ever ruled one out. In two dimensions the question was settled long ago: fluids always stay smooth. In our three dimensions, it is unknown.

What is actually being asked

In three dimensions, with smooth starting conditions and no external forces, prove that solutions to the incompressible Navier–Stokes equations exist for all time and stay smooth — or exhibit a solution that fails.

The prize is symmetric: a counterexample earns it just as surely as a proof. Note what is not being asked. We already know solutions exist for a short time, and we know weaker “weak solutions” exist forever. The gap is between weak-forever and smooth-forever.

Why anyone cares

Turbulence is the last great unsolved problem of classical physics, and this is its precise mathematical core. Every simulation on Earth — aerodynamics, climate models, blood-flow surgery planning — quietly assumes the solution it is computing actually exists. If blow-ups are possible, the equations stop describing reality at some scale, and we would need to know where.

Where it stands

Open, and moving faster than any other problem on this page. Leray built the weak solutions in 1934. The 2D case is fully understood. In 2016 Terence Tao constructed a blow-up for a carefully averaged version of the equations, showing that any proof of smoothness must use more than energy bookkeeping — and hinting that the true answer might be blow-up after all.

That hint now looks prophetic. Beginning with Córdoba and Martínez-Zoroa's 2024 construction for a simpler fluid equation, a “multiscale” technique has been carried step by step up the ladder of harder equations — porous media, then Boussinesq, then three-dimensional Euler — and in September 2026 it reached Navier–Stokes itself, in the forced form described above. Tao, writing about the human half of this work, judged that completing the programme now “looks very feasible in the near future”.

So the honest summary is this: the unforced problem is still unsolved, the forced problem has a claimed and machine-checked proof that the community has not yet digested, and this is the most likely of the six to fall next.

Try it → Drag inside the box to stir the fluid. This is the real thing — a solver for these very equations. Watch the peak vorticity readout: in this 2D world it is a theorem that it stays finite. Add a third dimension and it becomes a million-dollar question.

The everyday version

This is the hardest of the seven to picture, so start with an analogy. Imagine you can only build with Lego. Someone shows you a shadow on a wall and asks: can that shadow always be cast by something you built out of Lego bricks? Sometimes obviously yes. Sometimes it looks Lego-ish — right proportions, right angles — but you cannot find the model that casts it.

In Hodge's setting the “Lego bricks” are algebraic varieties: shapes carved out by polynomial equations. The “shadows” are topological features — the holes, loops and voids of a shape, catalogued by a machinery called cohomology.

Hodge discovered that on nice shapes this catalogue splits into neat compartments, and that one particular diagonal set of compartments looks suspiciously algebraic. His conjecture: those really are the shadows of Lego models. Every such class comes from an honest sub-shape cut out by polynomials.

It is a claim that a purely analytic smell always has an algebraic source. A bridge between three fields that were not supposed to touch.

What is actually being asked

On a non-singular complex projective variety, every Hodge class is a rational combination of the classes of algebraic cycles.

The diagram on the right is the “Hodge diamond”, the actual catalogue. Its diagonal is where Hodge classes live. The conjecture says nothing on that diagonal is a mirage.

Why anyone cares

Algebraic geometry has two ways of describing a shape: by writing equations, and by counting holes. They give different information and translating between them is one of the central difficulties of the subject. The Hodge conjecture would be the strongest dictionary yet, and it feeds directly into the theory of motives — a grand unification that much of modern number theory is waiting on.

Where it stands

Open, but not untouched. The first diagonal case is a theorem — Lefschetz proved it in 1924, before Hodge even stated the general conjecture. It is also known for every shape of three or fewer complex dimensions, and in scattered families beyond. Everything after that is conjecture, and there is no consensus that it is even true.

Try it → Explore the Hodge diamonds of real shapes and click the highlighted diagonal cells. Then switch tabs and draw a loop on a doughnut to see what a “class” actually is.

The everyday version

Take an elliptic curve, say y² = x³ − 25x. Now ask an ancient question: which points on it have coordinates that are ordinary fractions?

These curves have a magic trick. Draw a straight line through two fraction points; it hits the curve in exactly one more place, and that third point is a fraction point too. So from two solutions you can manufacture a third, then a fourth, forever. Sometimes this process cycles back on itself and the curve has only finitely many solutions. Sometimes it runs away and produces infinitely many, with numerators exploding to hundreds of digits.

Which happens? There is no known method guaranteed to tell you. Birch and Swinnerton-Dyer, running one of the first computers at Cambridge in the 1960s, tried something indirect: for each prime p, count the curve's solutions on a clock face of size p. Those counts are easy — finite arithmetic. And they noticed that curves with infinitely many fraction points tend to have more solutions than average on those clock faces, consistently, forever.

The whole conjecture is that this hunch is exactly right, and quantitatively so.

What is actually being asked

Bundle all the clock-face counts into one function, the L-function L(E, s). Bundle all the fraction solutions into one number, the rank.

The rank of E over the rationals equals the order of vanishing of L(E, s) at s = 1.

In words: the number of independent infinite families of solutions equals how flat the L-function is at one specific point. A count of things you can never finish, read off from a function built entirely out of finite counts.

Why anyone cares

Elliptic curves are the workhorses of modern number theory — Wiles used them to prove Fermat's Last Theorem — and they secure a large fraction of internet traffic through elliptic-curve cryptography. BSD is also the model case for a much larger web of conjectures linking L-functions to arithmetic. It even settles a problem from the tenth century: which whole numbers can be the area of a right triangle with fraction-length sides.

Where it stands

Open, with the shallow end conquered. If the L-function vanishes to order 0 or 1, the conjecture is a theorem, by work of Gross, Zagier and Kolyvagin in the late 1980s. Combined with later results this covers a majority of all elliptic curves. Rank 2 and above remains completely open, as does the refined version predicting the exact leading coefficient.

Try it → Watch the chord-and-tangent trick generate rational points with absurd numerators, then run the original 1960s experiment yourself — the point counts really do give the rank away.