In July 1986, Terence Tao represented Australia at the International Mathematical Olympiad in Warsaw. He was just 10 years old. He earned 19 out of 42 points and received a bronze medal, becoming the youngest IMO medalist. He earned full marks on Problems 1 and 2, although he received only one point on the final problem. Tao returned in 1987 to win silver with 40 points, and in 1988 he won gold at age 13, becoming the youngest gold medalist in IMO history.

A prime number that survives being erased from either end. 739397 → 39397 → 9397 → 397 → 97 → 7 739397 → 73939 → 7393 → 739 → 73 → 7 Every number shown is prime. Even more remarkable: 739397 is the largest prime with this property. Only 15 such primes exist.

A number that attracts almost every four-digit number. Start with 3524. Arrange its digits in descending and ascending order, then subtract. 5432 − 2345 = 3087 8730 − 0378 = 8352 8532 − 2358 = 6174 Now something remarkable: 7641 − 1467 = 6174 Every four-digit number containing at least two distinct digits eventually reaches 6174 by repeating this procedure, always within seven steps. A numerical curiosity discovered by Indian mathematician D. R. Kaprekar.

They are consecutive integers, and both are perfect powers. No other two integers greater than 1, each expressible as a square or higher power, are consecutive. Not even among infinitely many integers. This was conjectured in 1844 and finally proved in 2002.

A fraction of a degree can change an entire pattern. At approximately 137.507764°, the golden angle, given by 360°/φ², helps explain the remarkable spiral arrangements seen in sunflower heads.

Every possible arrangement of 199 remains prime.

In classical physics, a particle cannot cross an energy barrier higher than its total energy. Quantum mechanics tells a different story. In 1926, Erwin Schrödinger introduced an equation describing how quantum states evolve. One of its consequences is quantum tunneling: a particle can have a nonzero probability of appearing beyond a barrier it cannot classically overcome. This phenomenon helps explain radioactive decay and underlies important modern electronic devices.

A referee found Einstein's mistake: In June 1936, Albert Einstein and his collaborator Nathan Rosen submitted a manuscript to Physical Review. Its title was provocative: Do Gravitational Waves Exist? Twenty years earlier, Einstein had predicted gravitational waves as a consequence of general relativity. Now, however, his new calculations seemed to suggest that such waves could not exist. Einstein was challenging a prediction of his own theory. The journal's editor sent the manuscript to an anonymous referee, physicist Howard Percy Robertson. Robertson identified a serious problem in the mathematical argument. Einstein was furious. On July 27, 1936, he withdrew the manuscript, objecting to its having been shown to another specialist before publication. He sent the work to the Journal of the Franklin Institute instead. But the story took an extraordinary turn. Robertson, who was also Einstein's colleague at Princeton, subsequently helped him recognize how the apparent mathematical difficulty could be resolved. Einstein revised the paper, and its conclusion changed. When the corrected paper appeared in January 1937, Einstein included a note acknowledging that the original mathematical results had been interpreted incorrectly. He explicitly thanked Robertson for helping clarify the mistake. Nearly eight decades later, on September 14, 2015, LIGO made the first direct detection of gravitational waves, produced by the merger of two black holes. The prediction Einstein had once doubted had become an observed feature of the universe.

During the 1860s, James Clerk Maxwell unified electricity and magnetism mathematically. His equations implied that disturbances in electric and magnetic fields could travel through empty space as waves. His equations predicted waves travelling through empty space at a speed given by c = 1/√(μ₀ε₀) The calculated speed matched the known speed of light. Maxwell concluded that light itself was an electromagnetic wave. A mathematical theory had revealed the nature of light before electromagnetic waves were produced and detected in the laboratory.

We learn four fundamental arithmetic operations in school: Addition, subtraction, multiplication and division. But mathematician Martin Eichler once suggested a fifth: modular forms. In a recent interview, Fields Medalist Richard Borcherds revisits this remarkable idea. Modular forms appear in unexpected places across mathematics, from number theory to the symmetries of the enormous Monster group. Borcherds, who won the Fields Medal for proving the monstrous moonshine conjecture, calls them among the most magical objects in mathematics. Full Interview: youtu.be/lhE4DZWFQ1U?si=qcaz…