Stories of Great Problems
From Euclid’s ancient number world to open questions today, follow when each problem appeared, how long it resisted, and the first experiment you can try. No proof expertise is required to make a conjecture with a drawing, colors, divisors, or computation.
One great problem, three kinds of immersion
From a story to your own conjecture
Problem stories open curiosity. The problem workshop is the next place, where curiosity becomes a testable idea.
- 1 · DiscoverYou are hereMeet the questionLearn when it appeared and why it still holds people’s attention.
- 2 · ChallengeTest one caseMake your first observation in five minutes with a drawing, calculation, or colors.
- 3 · DevelopBuild on ideasPublish an observation and grow it through comparisons, conjectures, and counterexamples.
How would you like to meet your first problem?
Choose the shape of your curiosity, not a proof credential. Every route starts without login and opens a stable, shareable detail page.
The Inscribed Square — Does Every Loop Hide a Square?
Hunt for four square vertices on a deliberately wobbly loop.
Find the first crack in this problemHadwiger–Nelson — How Many Colors Can Guard the Entire Plane?
Color unit-distance points and catch the moment a fourth color becomes unavoidable.
Find the first crack in this problemThe Perfect Cuboid — Can All Seven Lengths Be Integers?
Check an Euler brick’s six integers and its failing seventh length.
Find the first crack in this problemErdős #1196 — From an AI-Suggested Method to a Collaborative Proof
How large can the Erdős sum of a primitive set containing only large integers be? In 2026, eight authors checked and developed a method suggested by GPT-5.4 Pro output into a proof of a 1960s conjecture.
Collatz Conjecture (3n+1) — A Child Can State It, Yet No One Has Solved It
Start with any positive integer. If even, halve it; if odd, multiply by three and add one. Does every start eventually reach 1? Large computations and strong density results still fall short of a proof for every integer.
Riemann Hypothesis — The Enduring Question in a Six-Page Manuscript
Do all nontrivial zeros of the zeta function have real part one-half? In a six-page manuscript of 1859, Riemann wrote that after a few unsuccessful attempts he had set aside the search for a rigorous proof. The short claim became a vast programme for measuring irregularity among the primes.
P vs NP — Is Fast Verification the Same as Fast Search?
If a proposed solution to a decision problem can be verified in polynomial time, can a solution also be found in polynomial time? Familiar puzzles and routing tasks lead to a central question in complexity theory.
Twin Prime Conjecture — Are There Infinitely Many Close-Packed Prime Pairs?
Are there infinitely many pairs (p, p+2) where both are prime? The case is included in de Polignac’s 1846 conjecture. In 2013 Yitang Zhang proved the first finite upper bound for infinitely recurring consecutive-prime gaps, without reaching gap two.
Goldbach Conjecture — Is Every Even Number a Sum of Two Primes?
4 = 2+2, 6 = 3+3, 100 = 3+97. The modern form emerged from Goldbach and Euler’s 1742 correspondence; huge computation and the proof of the weak conjecture still do not settle the strong version.
Fermat's Last Theorem — A 358-Year Margin
Fermat claimed a marvelous proof that would not fit in the margin; the note became public only after his death, and no proof survived. Wiles announced a result in 1993, repaired its gap with Richard Taylor in 1994, and published the two papers in 1995.
Poincaré Conjecture — Can Loops Identify a 3-D Space?
If every loop in a closed 3-manifold can shrink to a point, must the space be the 3-sphere? Poincaré’s question was solved through Hamilton’s Ricci-flow program, Perelman’s three preprints, and detailed verification by several teams.
Moving Sofa Problem — A Claimed Optimality Proof Under Review
What is the largest planar shape that can turn a right-angle corner in a unit-width corridor? In 2024 Jineon Baek posted a 119-page preprint claiming Gerver’s shape is optimal. It is a compelling proposed resolution still undergoing scrutiny.
Continuum Hypothesis — Proven Independent of ZFC
"Is there an infinity between the naturals and the reals?" Conjectured by Cantor in 1878. Gödel’s CH direction (1940) and Cohen’s ¬CH direction (1963) together showed that standard ZFC decides neither side.
Navier-Stokes Equations — Do Smooth Solutions for Fluid Flow Always Exist?
"We use these equations daily to design airplane wings — yet we don't even know if smooth solutions always exist." Equations from 1822, open for 200 years.
BSD Conjecture — How Integer Solutions on Elliptic Curves Distribute
"Are rational points on an elliptic curve finite or infinite? The L-function knows." Conjectured 1965, open for 60 years.
The Inscribed Square — Does Every Loop Hide a Square?
Open for 115 years since 1911. Draw any wobbly loop. Must four points on that very line always form an exact square?
Hadwiger–Nelson — How Many Colors Can Guard the Entire Plane?
Open for 76 years since 1950. Points exactly one unit apart must have different colors. The answer is still one of only 5, 6, or 7.
Odd Perfect Numbers — Not One Found in 388 Years
and . Perfect numbers rebuild themselves from their divisors. But can one be odd? The explicit question has resisted since 1638.
The Perfect Cuboid — Can All Seven Lengths Be Integers?
Three edges, three face diagonals, one space diagonal. Make all seven integers. No such box has been found in 132 years since the published 1894 challenge.