Problem
A Spatial Generalization of the Pythagorean Theorem
De Gua’s theorem extends the Pythagorean theorem to the face areas of a right tetrahedron; a short proof uses orthogonal projections.
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Articles, problems and notes on mathematics. Browse by topic or article type.
Instrument
The Marking Gauge: From Ancient Layout Lines to Precision Measurement
A marking gauge turns parallel-line geometry into practical layout work, from woodworking to modern height measurement.
Problem
Archimedes' Arbelos: A Geometrical Masterpiece
Archimedes’ arbelos, bounded by three semicircles, reveals striking relationships between areas, arcs and straight lines.
Essay
Trémaux's Algorithm: An Ariadne's Thread for Any Maze
Trémaux’s maze algorithm, presented by Édouard Lucas as a mathematical Ariadne’s thread, connects systematic path marking with modern graph search.
Problem
Ozanam's Dinner Puzzle: Seven Friends and Seventeenth-Century Combinatorics
Ozanam’s seven friends want to try every dinner seating. The count depends on whether seats form a row or a circle and whether mirror images are distinguished.
Problem
Diophantus' Puzzle: Dividing the Number 100
An ancient problem from Diophantus asks for two divisions of 100 whose larger and smaller parts satisfy two ratios.
Essay
Infinite Descent: Mathematical Induction in Reverse
Fermat's method of infinite descent proves impossibility by constructing an endlessly decreasing sequence of natural numbers.
Essay
Kolmogorov’s Button Problem: A Great Mathematician’s Combinatorial Game
“Real mathematics should always be a little like a game: that is its beauty and strength.” — A. N. Kolmogorov
Essay
Allotrion: A Forgotten Word in the History of Science
Allotrion, a term for something alien to a subject, travelled from ancient logic into the teaching of mathematics.
Problem
Einstein’s Puzzle: The Geometry of Numbers
Among the many mathematical puzzles attributed to Albert Einstein, the problem of nine numbers and seven triangles has a special place.
Instrument
The Field Compass
An old measuring instrument that brings together mathematics, everyday life and peasant ingenuity: the field compass.
Essay
The Rule of False Position
The ancient rule of false position used trial values and proportional corrections to solve equations long before formal symbolic algebra.
Essay
The Knight's Tour: A Puzzle That Leapt Across the Centuries
From Euler to Warnsdorff’s rule, the knight’s tour connects a familiar chess puzzle with Hamiltonian paths and algorithmic search.
Problem
Clever Soldiers at an Inn
An old recreational problem by Ye. I. Ignatyev leads from soldiers’ dinner bill to the Josephus recurrence.
Problem
The Pythagorean Number Circle: Mysticism and Mathematics
Arrange numbers in a circle, count up and then down, and a square appears. An old puzzle from Yemelyan Ignatiev's book leads to a simple proof.
Essay
Raphael's School of Athens: Art, Philosophy and Science in Dialogue
Raphael's School of Athens brings ancient thinkers into a Renaissance vision of dialogue, mathematics and the harmony of art, philosophy and science.
Problem
Muhammad's Signature: A Geometrical Puzzle Between Science and Faith
Through geometry the soul touches the divine. Omar Khayyam
Essay
The Mouseion: The Scientific Olympus of the Ancient World
Alexander the Great’s conquests opened a new age in the ancient world, creating an empire larger than any that had existed before.
Essay
Zeno of Elea’s Paradoxes
Since ancient times, philosophers have used the categories of motion and rest, continuity and discreteness, and the infinite and finite to understand the world.
Problem
Three Brothers and Eight Écus: A Seventeenth-Century French Puzzle
Three brothers take turns giving money to one another and finish with eight écus each. Work backwards or use algebra to discover how much they started with.
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