Why does a plastic shower curtain often billow inward toward the water spray and stick to your legs while you shower, instead of staying put or blowing outward?
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Euler's polyhedron formula
Medium
A convex polyhedron has vertices, edges and faces. Prove that .
Contrary to the classic cartoon teardrop symbol, what shape does a falling raindrop actually have, and why isn't it pointed at the top?
Why does burning firewood make crackling, popping, and hissing sounds, and why is the noise so much louder and more frequent with some pieces of wood than others?
Pick's theorem
Medium
A simple polygon has all of its vertices at points with integer coordinates. It has integer points in its interior and integer points on its boundary. Find a formula for its area in terms of and , and prove it.
On a clear day, why does the sky directly overhead look a deep, saturated blue while the sky near the horizon looks pale, washed-out blue or nearly white?
Achilles, though ten times faster than a tortoise with a head start, seems logically unable to ever catch it, since by the time he reaches where the tortoise was, it has always moved a little further ahead.
The birthday problem
Easy
Assume birthdays are independent and uniformly distributed over 365 days. What is the smallest number of people for which the probability that at least two of them share a birthday exceeds 1/2?
When several bodies move relative to one another (e.g., a person walking inside a moving train, or a boat crossing a flowing river), why does naively adding or subtracting the given speeds as plain numbers often give an obviously wrong resulting speed or direction?
If you travel the first half of a trip's distance at 60 km/h and the second half at 30 km/h, is your average speed for the whole trip the simple arithmetic mean, 45 km/h?
The problem of points
Medium
Two players of equal skill play a series of fair rounds, and the first to win 6 rounds takes the whole stake. The game is interrupted when one player leads 5 rounds to 3. How should the stake be divided fairly?
A subway rider who travels at random times notices that trains heading in his preferred direction seem to arrive less often than trains going the other way, even though the timetable runs both directions with exactly equal frequency — how can this be?
Inside an accelerating vehicle, a ball resting on a frictionless floor suddenly starts rolling backward even though nothing visibly pushes it — apparently violating Newton's first law that a body at rest stays at rest unless acted on by a force.
A finite set of points in the plane does not lie on a single line. Prove that some line passes through exactly two of the points.
Why does the water pressure, and hence the total force, on the bottom of a container depend only on the height of the water column and the base area — not on the container's shape or the total weight of water inside it — so that a narrow vase can press on its base just as hard as, or harder than, a…
When two lumps of soft clay collide head-on and stick together, coming to a complete stop with no visible flash, sound, or motion afterward, where did all the kinetic energy go, if energy is supposed to be conserved?
guests check their hats, and the hats are handed back in a uniformly random order. Let be the probability that no guest gets their own hat. Find a formula for and its limit as .
Why does the free end of a chain falling off a table, or piling onto the floor, accelerate downward faster than gravity alone (g) would predict?
Why do raindrops in a storm follow a characteristic size distribution -- many small drops, few large ones -- as described empirically since the 1940s?
Fifteen schoolgirls go for a walk every day for a week, in five rows of three. Arrange the rows for the seven days so that no two girls walk in the same row more than once.