A set that shifts into its complement
Can arbitrarily small translations turn a subset of the real line into its complement?
real analysislinear algebraset theory
Problems
Take your time with a question. When a solution is included, it stays tucked away until you want to see it.
Cutting a cube
How many straight cuts does it take to separate a cube into 27 unit cubes, even if you can rearrange the pieces?
geometrypuzzles
Arranging an invertible matrix
Arrange any n² distinct real numbers in a square matrix with nonzero determinant.
linear algebradeterminantsinduction
Two expressions that cannot both be cubes
For a natural number n, prove that n + 2 and n² + n + 1 cannot both be perfect cubes.
number theoryperfect powers
Finding an invisible spaceship
Can a fixed collection of detectors distinguish every possible 2 × 2 spaceship position on a 7 × 7 board?
combinatoricspuzzles
The average distance across a circle
Choose two points uniformly on a unit circle. What is their expected straight-line distance?
probabilitygeometryexpectation
A three-variable inequality
Prove a symmetric inequality for arbitrary real x, y, and z.
algebrainequalities
Trigonometry without a word
Two arctangent identities, seen through a geometric proof without words.
geometrytrigonometry
Stirling’s approximation, but weaker
Compare factorial and exponential growth by proving a limit.
analysissequences
Simple but fun
A permutation of an odd-sized set produces an even product. Can you see why?
number theoryparity
Hilbert matrices
A deceptively simple matrix: can you prove it is always nonsingular?
linear algebraanalysis
Conservative polynomials
What happens to a polynomial when we take its absolute value?
algebrapolynomials