Hilbert's third problem
Very hard
A regular tetrahedron and a cube have the same volume. Can the tetrahedron be cut into finitely many polyhedral pieces that can be reassembled to form the cube? Prove your answer.
All problems
Post a problemHow does a pop-pop boat, which only pulses water in and out of a rear nozzle with no net average flow, manage to move forward continuously?
100 prisoners are put in separate cells and taken one at a time, in an order chosen by the warden, to a room with a light switch that is initially off. Every prisoner will be brought to the room again and again, but no one learns anything except the state of the switch. On each visit a prisoner may…
Why does a thin, fast vertical jet of water shoot upward after the air cavity created by a falling ball or drop collapses beneath the surface?
Buffon's needle
Medium
A floor is ruled with parallel lines a distance apart. A needle of length is dropped at random. What is the probability that it lands across one of the lines?
Why, when a container of mixed different-sized granular particles (like nuts or cereal) is shaken, do the largest particles reliably rise to the top rather than settling based on density?
A closed knight's tour on an board is a sequence of knight moves that visits every square exactly once and ends a knight's move away from the starting square. For which does a closed tour exist? As a first step, prove that none exists when and are both odd.
What is the physical mechanism that produces ball lightning — the luminous, floating, sometimes room-sized spheres of light occasionally reported during thunderstorms — and can it be reliably reproduced in the lab?
On a game show there are three doors, one hiding a car and two hiding goats. You pick a door. The host, who knows where the car is, always opens one of the other two doors to show a goat and then offers you the chance to switch to the remaining closed door. Should you switch, and what is your…
Why does life on Earth use almost exclusively 'left-handed' amino acids and 'right-handed' sugars, when the same basic chemistry could in principle work equally well with the mirror-image molecules, and what process originally broke this molecular symmetry?
The Tower of Hanoi
Easy
disks of different sizes are stacked on one of three pegs, largest at the bottom. A move takes the top disk from one peg and places it on another peg, never on top of a smaller disk. What is the minimum number of moves needed to transfer the whole stack to another peg? Prove your answer.
How do migratory animals like birds, sea turtles, and some fish sense Earth's magnetic field well enough to navigate across continents or oceans, and does this rely on a genuinely quantum-mechanical process inside a light-sensitive protein in their eyes?
The Josephus problem
Medium
people numbered to stand in a circle. Going around the circle starting from person 1, every second person is removed (first person 2, then 4, and so on, continuing around the circle) until one person remains. Find a formula for the position of the survivor.
A person wearing perfume enters a lecture hall; roughly how long does it take before someone across the room can smell it, and does ordinary molecular diffusion explain the timescale?
Liquid nitrogen is around -196 C, yet a small splash on bare skin often causes no injury at all: what protects the skin, and when does that protection fail?
Langford's problem
Hard
Arrange the numbers in a row so that for each exactly other numbers stand between the two copies of . For the sequence 2, 3, 1, 2, 1, 3 works. For which is such an arrangement possible?
Why does dragging a moistened finger around the rim of a wine glass produce a sustained musical tone, and why does adding water to the glass change the pitch?
Why does a droplet of water dropped onto a very hot skillet (well above boiling point) form a levitating bead that skitters around for a surprisingly long time instead of flattening and evaporating immediately?
100 prisoners numbered 1 to 100 face a room with 100 drawers, into which the numbers 1 to 100 have been placed in random order. One at a time, each prisoner enters, opens at most 50 drawers looking for their own number, and leaves the room exactly as it was, with no communication afterwards. All are…
Why does a spilled drop of coffee (or red wine) dry into a dark ring around the outside with a pale center, instead of leaving an evenly colored stain?