How the number 15 solved a longstanding math challenge

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.

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.

Figuring out how to get a sofa around a corner has been a math problem since 1966

A sofa with the largest possible area that can make a 90-degree turn is known as Gerver's sofa, named after the mathematician who determined the shape in 1992. In 2025, a 119-page paper provided the proof confirming this shape.

A 14-year-old "human calculator" shatters six mathematical world records in one day

A teenage prodigy from Maharashtra, India named Aaryan Shukla set the record for the fastest time to mentally add 50 five-digit numbers. He did so in just 25.19 seconds–roughly equivalent to one addition every 0.5 seconds. The math whiz said he practices calculations for around five to six hours a day.

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...

In mathematics, a 'perfect number' is equal to the sum of the numbers it is divisible by, excluding itself, such as 28, which can be evenly divided by 1, 2, 4, 7, and 14, and 1 + 2 + 4 + 7 + 14 equals 28.

Explore over 30,000 Olympiad-level math problems from around the world.

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.

How algorithmic redistricting works

Computational methods are convincing a number of judges that some state district maps are drawn illegally. Here's a simple overview of how it works.

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.

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.

Traffic flow can be improved through research into card shuffling

Card-shuffling mathematics has been applied to solving problems in biology to estimate the evolutionary distance between organisms and in computer science to determine the optimal distribution of files and folders in databases.

The Koch snowflake has an infinite perimeter enclosing a finite area

Counterintuitively, although it would be impossible to build a fence along the edges of the shape because it would require an infinite amount of materials, this perimeter can be contained in a larger shape with a finite perimeter and limited area. This is equivalent to enclosing an infinitely long spiral that never reaches the center within a circle.

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.

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.

Google’s quantum computer completes a 10,000-year task in about 3 minutes

Google used its 53-qubit processor on an extremely difficult problem for traditional computers to simulate to showcase quantum supremacy. IBM later challenged Google’s claim by arguing that its Summit supercomputer could solve the same problem in just 2.5 days using a more efficient method.

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.

Infinity can come in different sizes and be ordered by these sizes

Russian mathematician Georg Cantor demonstrated that certain sets of numbers (e.g., the natural numbers) form countable infinities, while others cannot be counted (e.g., the real numbers) and are therefore uncountable, representing larger versions of infinity. Through the axiom of choice, German mathematician Ernst Zermelo proved this concept of ordering infinities.

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.

In a standard deck of 52 cards, shuffling produces one of approximately 8.07 × 10⁶⁷ potential card orders—a number about as large as the number of atoms in the Milky Way.

Try counting like the Maya in this interactive math game

The Maya numerical system used dots for 1, bars for 5, and a shell symbol for zero. This interactive resource lets users practice counting with the Maya base-20 system using the same symbols once common in the Yucatán.

Test your gerrymandering skills with this interactive game

This simulation demonstrates how packing and cracking alter election outcomes. See how shifting district lines can change representation—regardless of the overall makeup of the electorate.

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.

The odds of a perfect March Madness bracket are worse than finding one specific grain of sand on Earth

With 63 games to predict, random guessing produces odds of 1 in 9.2 quintillion. Even an informed picker who gets 67% of games right faces odds of about 1 in 120 billion, meaning if every American filled out a unique bracket, we'd expect a perfect one in roughly 366 years.

Data suggests that claims about the golden ratio underlying the geometry seen in nature, from mollusk shells to human proportions, are largely exaggerated.

How March Madness bracketology became a national obsession

An estimated 60 million people fill out March Madness brackets yearly, despite a one in 9.2 quintillion chance of filling it out perfectly. This Smithsonian magazine article explores how UCLA's dominance in the 1960s and '70s made brackets boring and predictable, and how the expansion to 64 teams and putting games on TV played a huge role in popularizing March Madness brackets.

As of 2025, the College Board has banned the use of calculators with computer algebra systems on the SAT and PSAT.

When you complete a CAPTCHA on a website to prove you are human, you may be unknowingly training algorithms for self-driving cars, whose systems require training data to help identify crosswalks, bicycles, and other outdoor objects.

'Scorigami' attempts to chart every NFL game score

The rules of the sport set bounds on possible final outcomes, so this adept statistician mined some fascinating takeaways. Some scores are impossible (like 5 to 1) and some are more common (like 20-17, occurring over 300 times since 1944). Watch for a deluge of numbers.

How math whizzes exploit sportsbooks

Chris Dirkus is a professional sports bettor who approaches the hobby as a math problem, not from a fan's perspective. He focuses on mispriced odds in popular games where many fans are betting for high liquidity. He's one of a rare segment of sports bettors exploiting slow models and human mistakes.

Cooling silica nanoparticles lets physicists push the boundary of how large an object can still behave quantum mechanically.

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.

This flat map is still wrong, but it's the least wrong of all flat maps

While flat maps can never be as perfectly scaled as a globe, they're far more practical than carrying around a sphere. Researchers set out to minimize six major errors of flat maps, including boundary cuts and skewed proportions, by simply squashing...

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.

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.

Purdue students' robot solves a Rubik's cube faster than most people can blink

The solution time of 0.103 seconds is a Guinness World Record. A robot created by Mitsubishi Electric engineers held the previous record of 0.305 seconds.

Explore a visual guide to how quantum computers can crack RSA encryption

In 1994, Peter Shor introduced an algorithm that showed a quantum computer could efficiently factor the product of two very large prime numbers, undermining the mathematics that makes RSA so secure. While using larger keys can delay the decryption time of quantum computers, this solution increases the time it takes to encrypt data, which may become impractical.

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.

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.

The sophisticated math jokes of 'The Simpsons'

"The Simpsons" has always been a slyly sophisticated show and has featured more than a few references to complex mathematical concepts. That's largely due to its writers' room, which has featured more than one writer who studied math before transitioning to comedy.

Five charts that changed the world

Data visualization helps us understand the world, providing a powerful way to communicate complicated information in a quick and efficient way. There are many different ways to create graphs, and these five examples—the pie chart, geographic dot...

Visualize how data becomes organized through a variety of sorting algorithms

This interactive shows how different algorithms shift elements in a randomized set to order them from smallest to largest. By comparing the different procedures and their corresponding functions, users can explore how runtimes vary across algorithms that complete the same task.

Tetrachromacy allows humans to observe 10 times the normal amount of colors

Resulting from a mutation on the X chromosome, the rare genetic condition causes individuals to possess an additional type of photoreceptor cell, typically sensitive to the orange range, located between the standard red and green cones. This enables the perception of approximately 100 million colors.

Claude Shannon, for whom Anthropic's AI tools are named, proved the existence of an unbreakable cipher

In the one-time pad approach, the encryption key is randomly generated and of equal length to the plaintext. This randomness ensures that every possible plaintext is equally likely—RBBHN can mean HELLO, SOLVE, CODES, and so on, using a different key—while matching the message and key lengths prevents the appearance of patterns that can be exploited.

The 15-minute city lets residents meet their daily needs within 15 minutes on foot

Many cities have been designed with inefficient layouts that waste time and lower the quality of life of those who live there. In the 15-minute city, essential services—including work, food, education, and health—should all be located within a short walk or bike ride from where people live.

Why patterns in art are so pleasing

Repeated visual cues like shapes or lines derive from the brain's innate compulsion to identify faces, places, bodies, and more. With repeated shapes, the brain seeks out resolution to the subtle variations between them. See examples in art and music.

How a brain injury created a mathematical mind

In 2002, two men savagely attacked Jason Padgett outside a karaoke bar, leaving him with a severe concussion--and turning Padgett into a mathematical genius.

Later in life, the clock in Grace Hopper's office ran counterclockwise

Acknowledging that most people have "an enormous tendency to resist change," Hopper's clock served as a reminder for the computer scientist that just because things have been done a certain way does not mean they should forever remain that way: "They love to say, 'We've always done it this way.' I try to fight that." (Some users may experience a paywall.)

The missing continent that took 375 years to find

It took scientists 375 years to discover the eighth continent of the world, which had been hiding in plain sight all along. But mysteries still remain.

Guess US city populations via bracket-style game

The US Census Bureau's data visualization team created a game—laid out like a March Madness basketball bracket—where you try to decide which metro area of two has the greater population. Test your knowledge of relative city populations and see...