Plot five points at random at the intersections of a coordinate grid. Between each pair of points a line segment can be drawn. Prove that the midpoint of at least one of these segments occurs at an intersection of grid lines.
By William Crane Jr., from the Sydney Town and Country Journal, 1877. White to mate in two moves.
A worm crawls along an elastic band that’s 1 meter long. It starts at one end and covers 1 centimeter per minute. Unfortunately, at the end of each minute the band is instantly and uniformly stretched by an additional meter. Heroically, the worm keeps its grip and continues crawling. Will it ever reach the far end?
By C.H. Wheeler, from the Dubuque Chess Journal, December 1877. “White to play and compel self-mate in two moves” — that is, White must force Black to checkmate him in two moves.
Let’s play a coin-flipping game. At stake is half the money in my pocket. If the coin comes up heads, you pay me that amount; if it comes up tails, I pay you.
Initially this looks like a bad deal for me. If the coin is fair, then on average we should expect equal numbers of heads and tails, and I’ll lose money steadily. Suppose I start with $100. If we flip heads and then tails, my bankroll will rise to $150 but then drop to $75. If we flip tails and then heads, then it will drop to $50 and then rise to $75. Either way, I’ve lost a quarter of my money after the first two flips.
Strangely, though, the game is fair: In the long run my winnings will exactly offset my losses. How can this be?
“The Forlorn Hope,” by J. Paul Taylor, 1878. White to mate in two moves.
Cries logical Bobby to Ned, will you dare
A bet, which has most legs, a mare, or no mare.
A mare, to be sure, replied Ned, with a grin,
And fifty I’ll lay, for I’m certain to win.
Quoth Bob, you have lost, sure as you are alive,
A mare has but four legs, and no mare has five.
– The Panorama of Wit, 1809
In a solar eclipse, the moon casts its shadow on Earth. In a lunar eclipse, Earth casts its shadow on the moon.
Solar eclipses are more common than lunar eclipses, but we tend to have the opposite impression. Why?
By Philip Hamilton Williams, Evening News and Post, 1895. White to mate in two moves.
Mount Everest rises 29,029 feet above sea level, and Ecuador’s volcano Chimborazo rises only 20,702 feet. But because Earth bulges at the equator, Chimborazo is actually farther from the center of the planet. If we could connect the two peaks with a water pipe, in which direction would the water flow?
Imagine two concentric roulette wheels, each divided into 100 sectors. Choose 50 sectors at random on each wheel, paint them black, and paint the rest white. Prove that we can now position the wheels so that at least 50 of the aligned sectors match.
A carnival worker is asked to paint the deck of a carousel. Because the center of the carousel is occupied by machinery, he can’t measure its diameter or even its radius. The best he can do is to take the measurement shown in green, which is 42 feet.
He’s explaining this apologetically when his supervisor stops him. “That’s all the information we need,” he says. “That’s enough to tell us how much paint to buy.”
How did they go about it?
Alice gets a rocket-powered pogo stick for her birthday. She jumps 1 foot on the first hop, 2 feet on the second, then 4, 8, and so on. This gets alarming. By judicious hopping, can she arrange to return to her starting point?
You’re alone on a desert island and want to lay out a course for some snail races. Unfortunately, you have only an 8.5 x 11 inch sheet of paper. How can you use it to measure exactly 3 inches?
You and a friend are playing a game. Between you is a pile of 15 pennies. You’ll take turns removing pennies from the pile — each of you, on his turn, can choose to remove 1, 2, or 3 pennies. The loser is the one who removes the last penny.
You go first. How should you play?
I release a fish at the edge of a circular pool. It swims 80 feet in a straight line and bumps into the wall. It turns 90 degrees, swims another 60 feet, and hits the wall again. How wide is the pool?
Fifty-five chameleons live on a tropical island. Thirteen are green, 19 are brown, and 23 are gray. Whenever two chameleons of different colors meet, both change to the third color. Is it possible that all 55 chameleons might eventually be the same color?
Fourteen ladders stand in a row. At the foot of each ladder is a monkey; at the top is a banana. Festooning the ladders are an arbitrary number of ropes. Each rope connects a rung on one ladder to a rung on another, but no rung receives more than one rope.
At a signal all 14 monkeys begin climbing. If a monkey encounters a rope it climbs along it to the other end and then continues climbing upward. Show that every monkey gets a banana.
You have 13 reels of magnetic tape, one empty reel, and a machine that will wind tape from a full reel to an empty one, reversing its direction. You need all 13 tapes reversed on their original reels. Show how this can be done, or prove that it’s impossible.