One great problem, three kinds of immersion
From a story to your own conjecture
Problem stories open curiosity. The problem workshop is the next place, where curiosity becomes a testable idea.
- 1 · DiscoverYou are hereMeet the questionLearn when it appeared and why it still holds people’s attention.
- 2 · ChallengeTest one caseMake your first observation in five minutes with a drawing, calculation, or colors.
- 3 · DevelopBuild on ideasPublish an observation and grow it through comparisons, conjectures, and counterexamples.
The Inscribed Square — Does Every Loop Hide a Square?
Open for 115 years since 1911. Draw any wobbly loop. Must four points on that very line always form an exact square?
Test small cases immediately with a drawing, divisors, or colored pencils.
The starting level measures how easily you can understand and test small cases. It is not the difficulty of a complete proof.
Jump to your first five minutesSee the puzzle visually
Distort the loop as much as you like. The question is whether at least one square always keeps all four vertices on the boundary.
Problem statement
Does every simple closed curve in the plane contain four points that are the vertices of a square?
The story of this puzzle
Otto Toeplitz published the question in 1911. A circle contains endlessly many squares, and even an egg-shaped curve seems to reveal one after a little sliding and turning. The trap appears when the loop becomes jagged: four distances and four right angles must line up at once.
This is a problem where observation comes before proof. Draw five different loops, move a transparent square template around each one, and look for four simultaneous contacts. You will keep succeeding. The instant you ask whether a loop can be designed to defeat every square, you are already exploring the edge of a 115-year-old problem.
Joshua Greene and Andrew Lobb proved a major 2021 result in the Annals of Mathematics: every smooth Jordan curve inscribes a rectangle of every chosen aspect ratio. Yet the case of an arbitrary continuous simple closed curve, which may be extremely rough, remains open. The mystery lives exactly between the easy picture and the wild boundary.
Try it yourself
Your first five minutes. Draw a circle, ellipse, rounded triangle, indented loop, and very jagged loop. Rotate and scale a paper square until all four corners touch. Which loop made the square hardest to find? Write that feature as your first conjecture.
Beyond MathVoyage
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