Three by Three

three by three puzzle

What’s the ratio between the areas of the two triangles?

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Invert the smaller triangle and it becomes clear that the ratio is 4 to 1.

June 17, 2013June 28, 2013 | Puzzles

Well Traveled

In Lord Dunsany’s Fourth Book of Jorkens, a member of the Billiards Club observes a book called On the Other Side of the Sun and says, “On the other side of the sun. I wonder what’s there.”

Jorkens, to everyone’s surprise, says, “I have been there.”

Terbut challenges this, but Jorkens insists he was on the other side of the sun six months ago. Terbut knows perfectly well that Jorkens was at the club six months ago, so he wagers £5 that Jorkens is wrong. Jorkens accepts.

“You have witnesses, I suppose,” says Terbut.

“Oh, yes,” says Jorkens.

“My first witness will be the hall-porter,” says Terbut. “And yours?”

“I am only calling one witness,” says Jorkens.

“Went with you to the other side of the sun?” asks Terbut.

“Oh, yes,” says Jorkens. “Six months ago.”

“And who is he?” asks Terbut.

Whom did Jorkens call?

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The Astronomer Royal. Six months ago we were all on the other side of the sun.

June 15, 2013June 21, 2013 | Puzzles

Black and White

shinkman chess problem

By William Anthony Shinkman. White to mate in two moves.

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After 1. Qh8! Black must move his knight, allowing 2. e6#. (If 1. … Nd4 or 1. … Ng7, then 2. Qa8#.) From the Ottawa Citizen, 1902.

June 14, 2013June 21, 2013 | Puzzles

Alcohol Problem

A bottle of fine wine normally improves with age for a while, but then goes bad. Consider, however, a bottle of EverBetter Wine, which continues to get better forever. When should we drink it?

— John L. Pollock, “How Do You Maximize Expectation Value?”, Noûs, September 1983

See The Devil’s Game.

June 11, 2013June 11, 2013 | Puzzles

Out of Sight

Several spherical planets of equal size are floating in space. The surface of each planet includes a region that is invisible from the other planets. Prove that the sum of these regions is equal to the surface area of one planet.

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Wrap the whole group in an imaginary film. The film takes the shape of parts of cylinders, parts of planes, and parts of planets. The parts of the planets that contact the film are precisely those that are invisible to the other planets; a planet floating inside the group is entirely visible. The invisible regions on the planets are bordered by spherical arcs, parallel two by two. If we translate all these arcs onto a single planet, they’ll partition it into several parts, each corresponding to the invisible region of some planet. Together these parts cover the planet completely. A Soviet problem from the 22nd International Mathematical Olympiad in 1981. Here’s the same idea in two dimensions.

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June 10, 2013June 14, 2013 | Puzzles

Bad Behavior

You and I drive from Los Angeles to Las Vegas in separate cars. We depart simultaneously, and you stay always ahead of me, dutifully driving the speed limit throughout the trip. Nonetheless I get ticketed for speeding. How?

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I create a gap between us by slowing down, then break the speed limit in catching up.

June 8, 2013June 14, 2013 | Puzzles

Black and White

deutsche schachzeitung chess problem

From Deutsche Schachzeitung, 1890. White to mate in two moves.

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With the waiting move 1. Kh2! White forces Black to abandon some vital defense: If the Nb3 moves, then 2. b4#. If 1. … Nb5, then 2. Qa6#. If the Nd6 moves elsewhere, then 2. Nb7#. If 1. … Bb5, then 2. Qc7#. If the black bishop moves elsewhere along the a6-f1 diagonal, then 2. Nxb3#. If the black bishop moves along the a2-g8 diagonal, then 2. Qa6#. Note that the white king’s move must be to h2 — any other move leaves him exposed to a check by Black.

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June 7, 2013June 14, 2013 | Puzzles

Animal Behavior

Martian snakes are elastic. If you take the tail of a Martian snake, and I take the head, and we pull in opposite directions, will there always be a part of the snake that doesn’t move?

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Yes. The top of this trapezoid represents the unstretched snake, and the bottom represents the stretched snake. Every point on the top corresponds to one on the bottom, as denoted by the green lines. Since at least one of these lines must be perpendicular to both top and bottom, there must be some point on the snake’s body that does not move.

June 4, 2013June 14, 2013 | Puzzles

Reverses

A Hungarian problem shortlisted for the 30th International Mathematical Olympiad, 1989:

Around a circular race track are n race cars, each at a different location. At a signal, each car chooses a direction and begins to drive at a constant speed that will take it around the course in 1 hour. When two cars meet, both reverse direction without loss of speed. Show that at some future moment all the cars will be at their original positions.

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Imagine that each car carries a flag, and that when two cars meet they exchange flags. Now each flag travels around the course with a constant direction and speed, and so after 1 hour each flag has returned to its starting position. Because each car reverses direction when it meets its neighbor, the order in which the cars are arranged around the track does not change. This means that the starting position has now been rotated, perhaps onto itself, and that after an integral number of hours the original position must be reached again. See Measured Steps.

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