A Tiny Plus Solves a Grid Game That Defied Easy Answers
A simple plus-shaped tactic solves a long-standing grid game across every board size, with computer checks formally certified in Lean.
Math
The Hidden Rule That Lets Everyone Trade Places in a Locked City
A new theorem shows when constrained swapping is possible, how long routes may be, and why some high-connection networks still hide traps.
Inside the 60,000-Line Proof of an Eight-Dimensional Packing
A formal proof of the eight-dimensional sphere-packing theorem reveals the hidden work—and limits—behind machine-checked mathematics.
Why Repeating a Quantum Test Can Make Perfect Cheating Possible
A mathematical surprise shows how repeated quantum tests can turn an impossible one-shot conversion into certain success.
The Number Rule That Hides Inside a Nonrepeating Mosaic
Aperiodic tiles can carry exact arithmetic: hidden lattice addresses and scaling rules reveal order without repetition.
The Math That Traces Hidden Fluid Paths Through the Brain
Mathematics and MRI team up to infer how tracer may move through brain tissue, while revealing why hidden flow remains uncertain.
The Ternary Digits That Decide Where a Fractal Curve Turns
A new analysis of the base-three Delange function reveals how ternary digits encode its dense peaks, valleys and infinite slopes.
Two Nine-Line Diagrams Match, but Their Algebra Tells Another Story
Two nine-line arrangements share every intersection pattern, yet hide different algebraic relations—while a related measure stays predictable.
A Graph-Theory Surprise Shrinks the Search for Rare Geometric Cases
A graph-theory result links sparse shared-neighbor patterns to smaller epsilon-nets, revealing a surprising path around a logarithmic cost.
How Disconnected Fractal Dust Can Cast a Solid Shadow in Every Direction
A random fractal can be disconnected yet project to a gap-free interval in every direction, including shadows and distances from any point.
A Random Walk’s Valleys Ignore How Far Each Step Goes
A random walk’s valley count can ignore jump lengths—but its stopping rule can transform the statistics entirely.