## What Are We?

A riddle by Jonathan Swift:

We are little airy creatures,

All of different voice and features:

One of us in glass is set,

One of us you’ll find in jet,

T’other you may see in tin,

And the fourth a box within;

If the fifth you should pursue,

It can never fly from you.

What are we?

## Growing Pains

A problem from the Soviet Mathematical Olympiad:

Two hundred students are arranged in 10 rows of 20 children. The shortest student in each column is identified, and the tallest of these is marked A. The tallest student in each row is identified, and the shortest of these is marked B. If A and B are different people, which is taller?

## Odd and Even

Put the integers 1, 2, 3, … *n* in any order and call them *a*_{1}, *a*_{2}, *a*_{3}, … *a*_{n}. Then form the product

*P* = (*a*_{1} – 1) × (*a*_{2} – 2) × (*a*_{3} – 3) … × (*a*_{n} – *n*).

Now: If *n* is odd, prove that *P* is even.

## All in the Family

Two men are brothers-in-law if one is married to the other’s sister. What is the largest possible group of men in which each is brother-in-law to each of the others?

## Sweet and Sour

An ant will always position itself so that it’s precisely twice as far from vinegar as from honey. If we put a dab of vinegar at A and a dab of honey at B and we release a troop of ants, what formation will they take up?

## “Paradox”

From *Miscellanea Curiosa: or, Entertainments for the Ingenious of Both Sexes*, January 1734:

One evening, as I walk’d to take the Air,

I chanc’d to overtake two Ladies fair;

Each by the Hand a lovely Boy did lead,

To whom in courteous Manner thus I said:

Ladies! so far oblige me as to shew

How near akin these Boys are unto you?

They, smiling, quickly made this dark Reply,

Sons to our Sons they are, we can’t deny:

Though it seem strange, they are our Husbands’ Brothers,

And likewise each is Uncle to the other:

They both begot, and born in Wedlock were,

And we their Mothers and Grandmothers are.

Now try if you this Mystery can declare.

## Countdown

Will a prime number ever appear in this series?

9

98

987

9876

98765

987654

9876543

98765432

987654321

9876543210

98765432109

987654321098

9876543210987

…

## Round Trip

A problem by Hungarian mathematician Laszlo Lovász:

A track has *n* arbitrarily spaced fuel depots. Each depot contains a quantity of gasoline; the total amount of gas is exactly enough to take us around the track once. Prove that, no matter how the gas is distributed, there will be a depot at which an empty car can fill up, proceed around the track picking up gas at each depot, and complete a full round trip back to its starting depot.

## Table and Tumblers

This problem originated in Russia, according to various sources, but no one’s sure precisely where:

Before you is a square table that can rotate freely. In each corner is a deep well, at the bottom of which is a tumbler that’s either upright or inverted. You can’t see the tumblers, but you can reach into the wells to feel their positions.

Periodically the table rotates and stops at random. After each stop, you can feel two of the tumblers and turn over either, both, or neither. If all four of the tumblers are in the same state — all upright or all inverted — then a bell sounds. Otherwise the table rotates again and you make another “move.”

Can you guarantee to ring the bell in a finite number of moves? If so, how?

## Pastorale

A logic puzzle by Lewis Carroll, July 2, 1893. What conclusion can be drawn from these premises?

- All who neither dance on tight-ropes nor eat penny-buns are old.
- Pigs that are liable to giddiness are treated with respect.
- A wise balloonist takes an umbrella with him.
- No one ought to lunch in public who looks ridiculous and eats penny-buns.
- Young creatures who go up in balloons are liable to giddiness.
- Fat creatures who look ridiculous may lunch in public, provided that they do not dance on tight-ropes.
- No wise creatures dance on tight-ropes if liable to giddiness.
- A pig looks ridiculous carrying an umbrella.
- All who do not dance on tight-ropes and who are treated with respect are fat.

## Black and White

Otto Wurzburg, first prize, eighth tourney, *Pittsburgh Gazette-Times*, June 10, 1917. White to mate in two.

## Stormy Weather

Stephen Barr observes that a pitched roof receives less rain per unit area than level ground does. This seems to mean that rain that falls at a slant will be less wetting than rain that falls vertically. Why isn’t this so?

## Horse Sense

From the U.K. Schools Mathematical Challenge, a multiple-choice competition for students ages 11-14:

Humphrey the horse at full stretch is hard to match. But that is just what you have to do: move one match to make another horse just like (i.e. congruent to) Humphrey. Which match must you move?

## Catbird Seat

A ladder is leaning against a tree. On the center rung is a pussycat. She must be a very determined pussycat, because she remains on that rung as we draw the foot of the ladder away from the tree until the ladder is lying flat on the ground. What path does the pussycat describe as she undergoes this indignity?

## Cut and Thrust

University of Toronto math professor Ed Barbeau can take a rectangular piece of paper and, using only a pair of scissors, produce the object pictured above. How?

## Sound Sense

Here is a class of a dozen boys, who, being called up to give their names were photographed by the instantaneous process just as each one was commencing to pronounce his own name. The twelve names were Oom, Alden, Eastman, Alfred, Arthur, Luke, Fletcher, Matthew, Theodore, Richard, Shirmer, and Hisswald. Now it would not seem possible to be able to give the correct name to each of the twelve boys, but if you practice the list over to each one, you will find it not a difficult task to locate the proper name for every one of the boys.

## Four Puzzles

- You’re playing bridge. Each of four players is dealt 13 cards. You and your partner find that between you you hold all 13 cards of one suit. Is this more or less likely than that the two of you hold no cards of one suit?
- As he left a restaurant, a man gave the cashier a card bearing the number 102004180. The cashier charged him nothing. Why?
- How can you position a marble on the floor of an empty room so that I can’t hit it with a baseball?
- Thrice what number is twice that number?

## Crime Scene

A tricky problem by Ernest Clement Mortimer. This position was reached after Black’s fourth move in a legal chess game. Can you reconstruct the game?