A nice mathematical puzzle, with a solution anyone can understand, is to determine the direction a bicycle went when you come upon its tracks. The answer involves thinking about tangent lines, geometric constraints and the bicycle’s steering mechanism.
It is an unexplained fact that objects with icosahedral symmetry occur in nature only at microscopic scales. Examples include quasicrystals, many viruses, the carbon-60 molecule, and some beautiful protozoa in the radiolarian family.
Can you combine simple observations and mathematical thinking to show that atoms exist?
The mathematics of knot theory says that a simple loop and a trefoil are fundamentally different knots. But is that all there is to the question?
The Menger Sponge, a well-studied fractal, was first described in the 1920s. The fractal is cube-like, yet its cross section is quite surprising. What happens when it is sliced on a diagonal plane?