In 2000, the Clay Mathematics Institute placed million-dollar bounties on seven unsolved problems in mathematics—the Millennium Prize Problems—in the hopes of spurring solutions.

Explore over 30,000 Olympiad-level math problems

Users can browse questions from 47 countries and 17 languages, sorted by topic across categories that include algebra, geometry, discrete mathematics, and number theory. Each problem is accompanied by at least one solution, with both authored in LaTeX that can be copied over to other platforms.

How a competition honoring a king's birthday uncovered history's first chaotic system

A committee established a contest for King Oscar II's 60th birthday, featuring four unsolved problems in mathematics, including one on proving the stability of a system of more than two objects held together by gravity. Focusing on three celestial objects, Henri Poincaré won the prize for his proof, which he later learned was incorrect and, upon correcting it, instead proved these systems to be unstable.

Non-Nobel Prize awards recognize topics outside of chemistry, physics and medicine

Established in 2002, the Abel Prize awards $750,000 to a mathematician for pioneering scientific achievements, while the $1M Turing Award honors contributions in computer science. Additional prizes in earth science, engineering, environmental science and policy, astrophysics, nanoscience and neuroscience have also been established.

How the number 15 solved a longstanding math challenge

The “packing coloring” problem asks how many numbers are needed to fill an infinite grid so that identical numbers never get too close to one another. A new computer-assisted proof finds a surprisingly straightforward answer—only 15. While...

As of 2025, the Standard Model has at least five unresolved problems

Among the issues with the most successful theory of the universe at subatomic scales is its inability to explain the extreme imbalance observed between visible matter and antimatter, since both should be roughly equal. The Standard Model also omits gravity and cannot account for the observed rate of cosmic expansion.

Mathematicians often solve problems using wordless mental processes

It's a common technique among expert mathematicians, where thinkers rely on intuition, blurry perceptions and subconscious work—no writing, no formulas. The state is distinct from daydreaming, and often relies on sudden illuminations stemming from extended periods of subconscious reflection.

The most bizarre numbers in the universe

Even for math-y types, some numbers are incomprehensible. This article dives into several kinds of numbers more complicated than even irrational numbers with endless decimals like "pi" or Euler's number: Noncomputable numbers. Read the bizarre...

Researchers identify a hidden instability pattern in simple ocean wave models

Stokes waves—a solution to the mathematics of water waves—were once assumed to be stable. Italian mathematicians have proven disturbances cause these waves to collapse and such collapses follow an infinite repeating pattern. To complete the proof, the team offered a $100 donation to solicit online help.

Turbulence has been called the most important unsolved problem in classical physics

Despite being the most common type of fluid flow in our world, scientists have struggled to model turbulent flow accurately. Though behaving unpredictably, understanding inertia and viscosity can help determine whether turbulence will occur.

Only allowing three recipients causes the Nobel Prize to overlook team discoveries

While science has grown more inclusive and collaborative, the Nobel Prize system still struggles with outdated practices, such as secrecy and a rigid limit on awardees. This restriction sidelines many deserving scientists, especially in complex projects like the human genome or gravitational waves.

Within mathematics, some infinities are bigger than others

While the concept of infinity refers to a countless set of items, these infinite sets can be compared to one another by matching an item from one set to an item in another, one by one. This analysis shows that certain sets must be larger than others because some infinite sets have leftover, unmatched items.

The Pius XI Medal—named after the pope who rebranded the Pontifical Academy of Sciences in 1936—has been awarded every two years to a promising scientist under 45.

The Navier-Stokes equations are fundamentally accurate but unsolved in 3D

The equations model fluids and have been shown to have solutions for all systems in two dimensions. Averaging and approximations let the equations be used in climate models, aerodynamics, and medicine, though their use in some systems produces nonsensical answers.

The story of two female programmers running the simulations behind chaos theory's discovery

Margaret Hamilton and Ellen Fetter worked on MIT's 800-pound Royal McBee LGP-30 computer, which was fed binary code on paper tape to perform computations. Among the computations performed were those involving equations describing the motion of gas in a circulating fluid for Edward Lorenz, widely considered the father of chaos theory.

Mathematical breakthroughs influenced Shakespeare’s stories

New ideas in math were transforming Europe at the time Shakespeare penned his plays, including newly popular conceptions of zero via philosophers Descartes and Leibniz.

Critics say Nobel Prizes need updating to reflect how science and society work today

The rules still favor individuals over teams and may reflect outdated views on who does science. In 2013, the Nobel Prize in physics went to two theorists, even though over 5,000 scientists helped confirm the Higgs boson.

Quantum computers harness weird physics to explore unsolvable problems

Classical computers manipulate bits as either 0 or 1, but quantum computers use qubits to hold combinations of both, enabling new algorithms. This new approach could one day help with breakthroughs in medicine, chemistry, and AI.

It took 70 years for computer scientists to determine how to cut a cake fairly

The cake-cutting problem explores how to divide something fairly, which is not solely about cutting evenly-sized pieces. A protocol that accounted for personal preferences (e.g., a corner piece) between just three people took 16 years to develop.

The history of the 10 axioms of Zermelo–Fraenkel set theory with choice, regarded as the most universal foundations of math

Like postulates in geometry—accepted truths that form the logical building blocks for formal proofs of more complex theorems—axioms cannot be proven, but set the underlying logic and rules for systems. The ZFC axioms resulted from painstaking work to identify the underlying rules of mathematics and are built on the premise of sets, or well-defined collections of elements.

There are more chess game variations than observable atoms in the universe

Mathematician Claude Shannon estimated there are 10 to the 120th power variations of chess games possible. Known as the "Shannon number," this is more than the estimated number of atoms in the cosmos.

Using scientific bets, Hawking challenged and highlighted major physics ideas

Hawking famously placed wagers on high-stakes questions in theoretical physics—such as the Higgs boson—for fun and to draw attention to science. When Hawking lost a bet about Cygnus X-1 being a black hole, he paid up with a magazine subscription.

The Majorana 1 chip may provide quantum computing using geometry-based qubits

Traditional quantum computers are incredibly fragile, with encoded information easily disrupted and requiring significant error correction systems. Microsoft’s approach involves storing data on a surface, which is more stable, but involves an elusive particle first theorized in 1938.

Efforts to find Majorana particles may finally lead to reliable quantum computers

Though the particle—which is its own antiparticle—remains theoretical, scientists from Microsoft to Google continue developing systems to mimic this property towards building error-resistant quantum computers. Scientists at UCSB have been attempting to get the particle using just pencil lead and Scotch tape.

Misunderstandings about how quantum computing works may have overhyped their ability

Quantum computers are often viewed as machines that can solve all problems instantly by trying every solution simultaneously. Instead, they amplify correct answers and cancel out wrong ones, which only works for specific kinds of problems with special structures.

The US Navy used The 'Harvard Calculator' to solve advanced physics problems

Designed in 1937 by Harvard graduate student Howard H. Aiken, the IBM Mark I was a groundbreaking machine used to solve ballistic trajectories and other mathematical problems whose solutions were needed for the design of the atomic bomb in World War II. The device was one of the first mechanical computers.

One reason Americans are bad at math

The president of Barnard College explains why we need to better understand the links between math anxiety, math avoidance, and the US’s dismal test scores.

Quantum computers use qubits to solve problems much faster than classical machines

Unlike regular computers, where each bit is either 0 or 1, quantum computers use qubits, which can simultaneously be both—allowing for massive parallel processing power. Google’s quantum processor once solved a problem in seconds that would have taken a supercomputer 47 years.

Why the nature of consciousness is such a difficult problem

David Chalmers is one of the leading philosophers of consciousness. In this video, he outlines some of the major questions in this field, including the "hard problem of consciousness," which seeks to find neural correlate for every conscious experience.

Albert Einstein's equations predicted black holes decades before their discovery

For nearly 50 years, scientists dismissed black holes as implausible oddities, despite solutions to the equations indicating their existence. But the 1970s discovery of intense X-rays from Cygnus X-1 and its massive hidden companion flipped the consensus.

How quantum computers can empower Shor's algorithm to break digital encryption

Secure communications largely rely on numeric keys generated by multiplying two enormous prime numbers together, and it is computationally impractical to determine these primes by trying to divide by every possible one. While Shor's algorithm can provide better guesses for this brute-force approach, it relies on a number that is difficult to determine on classical computers but much easier to calculate using the simultaneous computations enabled by quantum superposition.

The 2011 Nobel Prize in physics was awarded for the discovery of dark energy

Through the use of standard candles—objects whose brightness is well-known—astronomers can determine distances by recognizing that the same phenomena will dim in predictable ways with increasing distance. By observing the same type of bright stellar explosion at various locations, Saul Perlmutter, Brian Schmidt, and Adam Riess identified that the universe's expansion was increasing.

Sleeping on a problem does seem to help—in fact research published in 2026 suggests scientists can seed people's dreams with sound cues about a difficult puzzle during REM sleep and they'll be better able to solve it.

As of 2025, the most cited papers in science include a review of graphene, methods to improve image recognition, streamlining electron density calculations, and optimizing protein studies.

The mathematics of general relativity does not rule out the existence of wormholes

Such objects, also known as Einstein-Rosen bridges, would resemble funnel-like tunnels connecting two regions of spacetime, allowing for instant travel across the universe. However, such structures would be incredibly unstable and require undiscovered materials with negative energy to keep them open.

The mystery of the expanding universe

One of the most surprising discoveries about our universe over the past century is that it's expanding. Even more surprising is that it’s speeding up, and scientists aren’t entirely sure why. The rate should be roughly dependent on the total energy contained in the universe, but calculations keep coming up short. Explore the so-called “Hubble Tension,” along with potential candidates to solve the riddle.

It took a month for the Rubik's Cube inventor to solve the cube

The Rubik's Cube is the most popular puzzle toy in the world, with over 350 million sold since its invention in 1975. Originally created as a tool to teach three-dimensional movement, the colorful problem-solving game skyrocketed in popularity, spawning speedcubing competitions and becoming a fixture of global toy culture.

Einstein and the fourth dimension

How Einstein's greatest theoretical achievement contains a simple but mind-bending proposition—the dimension of time is physically connected to the three observable spatial dimensions.

As of 2026, the Pontifical Academy of Sciences, which advises the pope on scientific matters, has included 90 Nobel laureates among its members.