Millennium Prize Problems for Kids: Why Six Remain

On May 24, 2000, in Paris, the Clay Mathematics Institute named seven problems and attached one million dollars to each. Seven million dollars, seven questions. The Millennium Prize problems kids hear about in passing are usually reduced to “hard maths with money attached,” which misses the interesting part: in twenty-six years, exactly one has been paid out, and the man who earned it said no.

Grigori Perelman proved the Poincaré conjecture in 2002 and 2003. He declined the prize in 2010, stating that “Hamilton’s contribution was no less than his own.” Six remain open.

Key Takeaways

• Seven problems, announced May 24, 2000, with a $1 million prize each and a $7 million total fund.
• One is solved. Perelman’s proof of the Poincaré conjecture, per Clay’s own summary, “tells us that every three manifold is built from a set of standard pieces, each with one of eight well-understood geometries.”
• He turned the money down in 2010, saying Richard Hamilton’s contribution was no less than his own. That detail tells a kid something about mathematics that no equation does.
• Winning is deliberately slow. Solutions must appear “in the form of peer-reviewed publications, which would then be further scrutinised,” and Clay “only considers proposed solutions to Millennium Prize problems after at least two years since publication.”
• One of the six has a live claim. OpenAI posted a claimed resolution of Navier–Stokes on September 8, 2026; as of October 2026 it “awaits scientific validation,” and OpenAI said it does not intend to claim the prize.

The Seven, One Honest Sentence Each

Short, accurate, no mysticism.

Poincaré conjecture (solved). If a three-dimensional shape has no holes in a certain precise sense, it must be a sphere. Perelman proved it, and the proof gave a complete classification of three-dimensional shapes into standard pieces with one of eight geometries.

P versus NP. If a computer can quickly check an answer, must it also be able to quickly find one? Almost everyone expects the answer is no, and nobody can prove it.

Riemann hypothesis. The prime numbers thin out as you count upward, and there is a function, the Riemann zeta function, whose zeros control exactly how. The hypothesis says all the interesting zeros sit on a single line. If true, the primes are as regularly distributed as they could possibly be.

Navier–Stokes existence and smoothness. The equations describing fluid motion always have solutions for a short time, but do those solutions stay smooth forever in three dimensions, or can the velocity blow up to infinity in finite time?

Yang–Mills existence and mass gap. The quantum field theory physicists use to describe strong nuclear forces works superbly in practice, and nobody has shown it is mathematically well-defined, nor proved that it predicts particles with a minimum nonzero mass.

Hodge conjecture. Complicated geometric spaces have features you can detect with tools from topology. The conjecture says, in a precise setting, that those features always come from solutions to polynomial equations.

Birch and Swinnerton-Dyer conjecture. For certain curved equations called elliptic curves, the conjecture connects how many rational-number solutions exist to the behaviour of an associated function at a particular point.

Three of those seven are, in a loose sense, about whether structure we observe is really there: Riemann about primes, Hodge about shapes, Birch and Swinnerton-Dyer about solutions. Two are about whether our physics is mathematically legitimate: Navier–Stokes and Yang–Mills. One is about the limits of computation. And one is finished.

Millennium Prize Problems Kids Can Actually Feel

Most of the seven need years of training to state carefully. Two of them a child can experience directly, which makes them the ones worth teaching.

P versus NP, felt with a jigsaw puzzle. Hand your child a 100-piece jigsaw, assembled, and ask them to check it is correct. Thirty seconds. Then break it up and ask them to assemble it. Forty minutes. Same puzzle, same child, same information. The ratio between those two times is the P versus NP question, in the only form that matters to a human.

Now the stake, which is not abstract. Internet encryption rests on problems that are easy to check and believed hard to solve. If someone proved that fast checking implies fast solving, and produced the method, a great deal of modern security would stop working. This is why a mathematics problem has a bank’s attention.

Riemann, felt with a sieve. Write the numbers 1 to 100 in a ten-by-ten grid. Cross out multiples of 2, then 3, then 5, then 7. What survives are the primes, and the method is Eratosthenes’ sieve, about 2,200 years old.

Then count primes per decade: 1 to 10, 11 to 20, and so on. You get 4, 4, 2, 2, 3, 2, 2, 3, 2, 1. They thin out, but raggedly. The Riemann hypothesis is a statement about exactly how regular that raggedness is. A ten-year-old who has made the grid by hand has the right picture in their head.

The Seven at a Glance

Problem	Status	What it asks, in brief	Can a kid feel it?
Poincaré conjecture	Solved, 2002–2003	Is a hole-free 3D shape always a sphere?	Partly, with modelling clay
P versus NP	Open	Is finding an answer as easy as checking one?	Yes, with a jigsaw
Riemann hypothesis	Open	How regularly do the primes thin out?	Yes, with a sieve
Navier–Stokes	Claim awaiting validation	Do fluid solutions stay smooth forever?	Yes, with a tap
Yang–Mills mass gap	Open	Is our theory of nuclear forces mathematically sound?	No
Hodge conjecture	Open	Do detected geometric features come from equations?	No
Birch and Swinnerton-Dyer	Open	How many rational solutions does a curve have?	Partly, with graph paper

Four of seven have a household demonstration. That is a higher hit rate than most lists of hard problems, and it is the reason this topic works at a kitchen table.

How to Teach Your Kid About These Problems

Ages 5–8: Checking is easy, finding is hard

Materials: a jigsaw puzzle, or a word search, and a timer.

Do the jigsaw experiment described above. Assemble the puzzle yourself first, have them verify it, and time that. Then break it and time the assembly. Write both numbers on paper, large.

Say the sentence once: “Checking took thirty seconds. Finding took forty minutes. Nobody knows if that difference has to be true, and there’s a million-dollar prize for finding out.” Children find the money motivating and the question memorable, in that order, and that is fine.

For a variant with no puzzle: hide an object in a room, time how long it takes them to find it, then time how long it takes them to confirm it is there once told. Same structure, zero materials.

Ages 9–12: Build the sieve, count the primes

Materials: squared paper, a pencil, two coloured pencils.

Grid out 1 to 100. Cross out multiples of 2 in one colour, then 3, 5, and 7 in another. What is left are the primes below 100, and your child has just executed an algorithm from roughly 240 BC.

Then the counting exercise: primes per decade, written as a row of numbers. Ask whether the pattern is predictable. It is not, obviously, and that is the point. Explain that there is a precise claim about how irregular it is allowed to be, and that proving it is worth a million dollars.

If they want more, extend the grid to 200. The thinning becomes more obvious and the raggedness persists. Both observations are correct and both matter.

Ages 13+: Measure the gap, then write about it

Materials: five sudoku puzzles of the same difficulty, a timer.

For each puzzle, time two things: how long to solve it, and how long to verify a completed grid is valid. Record ten numbers. Compute the ratio for each puzzle and look at the spread.

Then a writing task, 200 words: what would change if solving were as easy as checking? The honest answer touches encryption, logistics, drug design, scheduling and code-breaking. A teenager who writes that paragraph understands why P versus NP is on the list, and they will have arrived there from a timer and a pencil rather than from a lecture.

The question to ask: “Why would a million dollars not be enough to make someone solve one of these?”

What to Do at Home

Use Perelman’s refusal as the real lesson

A man solved a problem nobody had solved in a century and turned down a million dollars because he thought the credit was shared unfairly. Whatever you think of that decision, it is a clearer statement about what mathematicians value than any amount of explaining. Tell your kid the story and let them argue about whether he was right.

Keep the two-year rule visible

Clay requires peer-reviewed publication and then at least two years before it will consider a submission. When a kid hears a claim about one of these problems, the useful response is “then we’ll know in about two years.” That reframes impatience as a feature of the system rather than a flaw in it.

Treat the Navier–Stokes claim as a live case study

This is unusual and worth using. A famous problem has a claim attached right now, with a formal proof file, a company announcement, a cautious institutional response and a priority dispute. Following it in real time with a teenager teaches more about how mathematics actually works than reading about Fermat’s Last Theorem after the fact.

Put the list on a wall

Seven lines of text. Status next to each. Cross one off, because one is done. Children respond to lists with one item crossed off in a way they do not respond to lists with none, and it quietly communicates that these are tasks rather than mysteries.

What not to do

Do not present these as the hardest problems in mathematics, because that is not quite what they are. They are seven problems a committee chose in 2000 as important, tractable-looking and central. Plenty of hard open problems are not on the list, and plenty of famous ones were deliberately left off. Treating the list as a ranking of difficulty teaches a kid something false about how mathematics is organised.

What to Watch For Over the Next 3 Months

• Week 4: Check the status of the Navier–Stokes claim. Has a preprint appeared? Has anybody outside the claiming company reported checking the formal proof? That is the first external milestone and it is easy to look up.
• Month 2 red flags: News coverage announcing that a Millennium problem has been “won.” The prize cannot be awarded without peer-reviewed publication plus a two-year wait, so any such headline in this window is wrong on the facts, and spotting that is a good media-literacy exercise.
• Month 3 self-check: Ask your kid to name the one that was solved and the one with a live claim. If they can say Poincaré and Navier–Stokes, and add that Perelman declined the money, they have the whole shape of the story.

Frequently Asked Questions

How many Millennium Prize problems are left?

Six. The Poincaré conjecture was solved by Grigori Perelman in 2002 and 2003 and is the only one resolved. As of October 2026 a claimed solution to Navier–Stokes awaits scientific validation, so it is not yet counted as solved.

Why did Perelman refuse a million dollars?

He declined in 2010, stating that “Hamilton’s contribution was no less than his own,” referring to Richard Hamilton’s work on Ricci flow, which Perelman’s proof built on. He had earlier declined the Fields Medal. The decision is usually read as a judgement about how credit is distributed in mathematics.

Can a teenager realistically solve one?

Realistically, no, and the reason is not talent. Each problem sits at the end of years of specialised background, and the open ones have absorbed decades of effort by the best specialists alive. A teenager can absolutely understand P versus NP and the prime-counting question well enough to care, which is the useful goal.

How does someone actually claim the prize?

By publishing in a refereed mathematics journal and waiting. Clay’s rules require solutions to appear “in the form of peer-reviewed publications, which would then be further scrutinised before the prize is awarded,” and the Institute will only consider a submission after at least two years from publication.

Is P versus NP the most important one?

It has the broadest practical consequences, because encryption, scheduling and optimisation all depend on the gap between finding and checking. Whether that makes it most important depends on what you value. Mathematicians often name the Riemann hypothesis instead, because so many other results are written assuming it is true.

Did the Navier–Stokes claim win the prize?

No. OpenAI stated it does “not intend to claim the Millennium Prize for this result,” and the claim is awaiting validation in any case. The correct description is a claimed, formally checked resolution of statements (C) and (D) of the problem, with verification still ahead of it.

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About the author

Ricky Flores is the founder of HiWave Makers and an electrical engineer with 15+ years of experience building consumer technology at Apple, Samsung, and Texas Instruments. He writes about how kids learn to build, think, and create in a tech-saturated world. Read more at hiwavemakers.com.

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Sources

1. Clay Mathematics Institute. “Millennium Problems.” https://www.claymath.org/millennium-problems/
2. Wikipedia contributors. (2026). “Millennium Prize Problems.” Wikipedia. https://en.wikipedia.org/wiki/Millennium_Prize_Problems
3. Clay Mathematics Institute. “Navier–Stokes Equation,” official problem description by Charles L. Fefferman. https://www.claymath.org/millennium/navier-stokes-equation/
4. Wikipedia contributors. “Navier–Stokes existence and smoothness.” Wikipedia. https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness
5. OpenAI. (2026, September 8). “Navier–Stokes solution.” https://openai.com/index/navier-stokes-solution/
6. Google DeepMind. (2024, July 25). “AI achieves silver-medal standard solving International Mathematical Olympiad problems.” https://deepmind.google/discover/blog/ai-solves-imo-problems-at-silver-medal-level/
7. Wikipedia contributors. (2026). “2026 in science.” Wikipedia. https://en.wikipedia.org/wiki/2026_in_science

Related reading on HiWave Makers: whether AI can do original mathematics, how mathematicians will verify the Navier–Stokes claim, and turbulence, the unsolved thing your kid can see.