Lagrange multipliers: follow the constraint
Explore the method of Lagrange multipliers for constrained optimization, with a visual perspective.
calculusoptimizationinformation theory
Explorations
Follow a question wherever it leads—with explanations, mathematics, code, pictures, and experiments.
From planar graphs to the five Platonic solids
Count vertices, edges, and faces; add a point at infinity; then discover why there are exactly five regular convex polyhedra.
geometrygraph theorytopology
Four equal regions: two bisections and a turn
Two easy bisections leave one imbalance. A quarter-turn suggests how to remove it—but we must account for the jumps in a point count.
geometrycountingpuzzles
Hex: a game with no middle ground
Build a connection, challenge the computer, and walk a coastline to discover why Hex can never end in a draw.
gamestopologygraph theory
Three utilities: from the plane to a mug
Connect three houses to gas, water, and electricity. The plane refuses—but a mug, a donut, and a square with joined edges reveal a way through.
topologygraph theorypuzzles
Lights Out: every light can be flipped
Play on a grid or a graph, reveal a solution, and discover why every pattern can be complemented—and why successful routes agree on parity.
combinatoricsgraph theoryparity
How big can a cevian triangle be?
Three lines meet. Three side points make a triangle. Move the meeting point—and discover why a quarter is the most you can get.
geometryinequalitiessimplexes
The magician’s problem
A coin, an audience, and a probability that seems to settle—until you magnify a ripple that never disappears.
probabilityoscillationslimits