## Paperwork

For 25 years, Macalester College mathematician Joe Konhauser offered a “problem of the week” to his students. Here’s a sample, from the collection *Which Way Did the Bicycle Go?* (1996):

Fifteen sheets of paper of various sizes and shapes lie on a desktop, covering it completely. The sheets may overlap one another and may even hang over the edge of the desktop. Prove that five of the sheets can be removed so that the remaining ten sheets cover at least two-thirds of the desktop.

## Even Steven

A Scholar traveyling, and having noe money, call’d at an Alehouse, and ask’d for a penny loafe, then gave his hostesse it againe, for a pot of ale; and having drunke it of, was going away. The woman demanded a penny of him. For what? saies he. Shee answers, for y

^{e}ale. Quoth hee, I gave you y^{e}loafe for it. Then, said she, pay for y^{e}loafe. Quoth hee, had you it not againe? which put y^{e}woman to anon plus, that y^{e}scholar went free away.

— John Ashton, *Humour, Wit, & Satire of the Seventeenth Century*, 1883

## Trial by Fire

You are grilling steaks for Genghis Khan. Your little grill can broil two steaks at a time, but Genghis is hungry and wants three. That’s a problem: It takes 4 minutes to grill each side of a steak, so you’ll spend 8 minutes grilling the first two steaks, then another 8 grilling the third. Sixteen minutes is a long time to keep a warlord waiting.

How can you improve your time while still cooking the steaks thoroughly? Genghis really likes his well done.

## Do-Si-Do

I’m not sure who originated this puzzle. Can the white queen force the black king onto square a1 of this 4 x 3 chessboard?

## Round and Round

Does the top of a rolling wheel move faster than the bottom? In his *Cyclopedia of Puzzles* (1914), Sam Loyd calls this “an old problem which has created a considerable discussion in the mechanical world.”

The rolling wheel retains its shape; it will arrive at its destination as a connected unit. This seems to imply that all of its parts are moving at the same speed. Yet the point in contact with the ground is moving not at all, while the top continuously overtakes it. Surely, then, the top is moving faster? “There is just enough of the mathematical and mechanical element in the make-up of the problem,” writes Loyd, “to provoke discussions from such as are well-up on these subjects.”

His answer: “The top of a wheel progresses exactly as fast as the bottom.” And, being Sam Loyd, he adds a wrinkle: “If the question referred to a mark on the tire the answer would be different, for the top is the highest point of the wheel and cannot revolve, for if it revolves the hundredth part of an inch it ceases to be the top.”

A second vexed wheel riddle; William James and Lewis Carroll consider related questions; a train moves simultaneously east and west.

## Meant to Be

In 2007 Gordon Bonnet was going through some genealogical records when he ran across a marriage record for Edward DeVere Stewart and Etta Grace Staggers.

“It was only when I was putting the names in my database that I noticed something odd about them. What is it?”

## Long Range

A puzzler from Willis Ernest Johnson, *Mathematical Geography*, 1907:

“A man was forty rods due east of a bear, his gun would carry only thirty rods, yet with no change of position he shot and killed the bear. Where on earth were they?”

## The Missing Footmen

Like Nabokov, Lord Dunsany was fond of composing chess problems. This example was published in Hubert Phillips’ *Week-end Problems Book* of 1932:

Two eccentric gentlemen abandoned this position at a chess club — White had announced mate in four. What was the mate?

The key is to notice that Black’s king and queen have changed positions. This is not possible if the black pawns are on their home squares. And this means that we’ve been viewing the position upside down:

Now it’s clear that Black is hemmed in by his own pawns — White can mate him in four moves with the b8 knight, e.g. by 1. Nc6 Nf3 2. Nb4 (threatening Nd3#) Ne5 3. Qxe5 (any) 4. Nd3#.

Dunsany drew a game against Capablanca in a simultaneous exhibition in London in April 1929. He later wrote:

One art they say is of no use;

The mellow evenings spent at chess,

The thrill, the triumph, and the truce

To every care, are valueless.

And yet, if all whose hopes were set

On harming man played chess instead,

We should have cities standing yet

Which now are dust upon the dead.

## Gender Issues

In a certain kingdom, boys and girls are born in strictly equal proportions. Determined to increase the proportion of women in the land, the sultan issues a decree: Any woman who bears a son is forbidden to have any further children. He reasons that some families will thus contain multiple daughters but a single son.

A number of years pass, and the sultan is confused to find that the kingdom still contains an equal number of boys and girls. Why?

## The Motor Hire

A puzzle from R.M. Abraham, *Diversions & Pastimes*, 1933:

Michael O’Bleary hired a motor-car at a cost of fifteen dollars to take him to Ballygoogly market and back again in the evening. When he got half-way on his outward journey he met a friend, gave him a lift to the market, and brought him back to the point where he picked him up in the morning. There was a dispute about the payment. How much should Michael charge his passenger for his share of the motor hire?

## Truel

You must participate in a three-way duel with two rivals. Each of you is given a pistol and unlimited ammunition. Unfortunately, you, Red, are the weakest shot — you hit your target only 1/3 of the time. Black is successful 2/3 of the time, and Gray hits everything he aims at.

It’s agreed that you will take turns: You’ll shoot first, then Black, then Gray, and you’ll continue in this order until one survivor remains. At whom should you shoot?

## Adding Up

What is the sum of all the figures in the numbers from 1 to 1 million?

Hint: With the right technique, this can be done in the head.

## Numbers Game

In 1938, Samuel Isaac Krieger of Chicago claimed he had disproved Fermat’s last theorem. He said he’d found a positive integer greater than 2 for which 1324^{n} + 731^{n} = 1961^{n} was true — but he refused to disclose it.

A *New York Times* reporter quickly showed that Krieger must be mistaken. How?

## Suicide

By Sam Loyd. In how few moves can White force Black to mate him?

The answer is to castle queenside:

Now every Black move is mate.

## Dotty

Each point in an infinite plane is colored either red or blue. Prove that there are two points of the same color that are exactly 1 meter apart.

## Don’t Shoot!

The

Schachfreund, edited by M. Alapin, gives the following amusing Chess Skit. A well-known chess master allowed weak opponents to make as many moves as they pleased during five minutes, as odds, before the beginning of a game, with the provision that they confined their moves to their own half of the board. At the end of the five minutes the game commenced, the odds-giver having the first move. During the five minutes one of them had played: [1. a4 2. Na3 3. h4 4. Nf3 5. d4 6. Nd2 7. Rh3 8. Nac4 9. Raa3 10. Ne4 11. Qd2 12. Rhf3 13. g3 14. Bh3 15. Qf4 16. Rae3], whereupon the odds-giver resigned without having made a single move, as he could not avoid mate in two.

— *The British Chess Magazine*, January 1899

## Sideline

In the 1890s an eminent Scot began to publish short popular science articles under an assumed name, for “the fun of seeing if he [could] make another reputation for himself.”

He succeeded, publishing three articles in the *National Geographic* before the secret leaked out.

The pseudonym was H.A. Largelamb. Who was the man?

## The Problem of the Calissons

A calisson is a flat French candy traditionally manufactured in the shape of two equilateral triangles joined along an edge. Suppose a quantity of these are packed randomly into a hexagonal container:

Each candy must take one of three orientations: east-west, northeast-southwest, or northwest-southeast.

As it happens, no matter how the candies are packed into the hexagon, an equal number will take each of these three orientations.

In the May 1989 issue of the *American Mathematical Monthly*, Guy David and Carlos Tomei demonstrated this with a beautifully intuitive “proof without words.” What had they seen?

## Lincoln Seeks Equality

You’re in a pitch-dark room. On a table before you are 12 pennies. You know that 5 are heads up and 7 are tails up, but you don’t know which are which. By moving and flipping the coins you must produce two piles with an equal number of heads in each pile. How can you do this without seeing the coins?

## Straight and Narrow

Draw three nonintersecting circles of different sizes, and bracket each pair of them with tangents. Each pair of tangents will intersect in a point, and these three points will always lie along a line.

On being shown this theorem, Cornell engineering professor John Edson Sweet paused and said, “Yes, that is perfectly self-evident.” What intuitive proof had he seen?

## All’s Fair

Literary critic A.N. Wilson panned Bevis Hillier’s 1988 biography of John Betjeman. To get even, Hillier forged a love letter ostensibly written by Betjeman in 1944 and forwarded it to Wilson under the name Eve de Harben (an anagram for “Ever been had?”). Wilson took the bait and included the forged letter in his own biography of Betjeman, which was published in 2005.

Here’s the letter. It contains a hidden message — can you find it?

Darling Honor,

I loved yesterday. All day, I’ve thought of nothing else. No other love I’ve had means so much. Was it just an aberration on your part, or will you meet me at Mrs Holmes’s again — say on Saturday? I won’t be able to sleep until I have your answer.

Love has given me a miss for so long, and now this miracle has happened. Sex is a part of it, of course, but I have a Romaunt of the Rose feeling about it too. On Saturday we could have lunch at Fortt’s, then go back to Mrs. H’s. Never mind if you can’t make it then. I am free on Sunday too or Sunday week. Signal me tomorrow as to whether and when you can come.

Anthony Powell has written to me, and mentions you admiringly. Some of his comments about the Army are v funny. He’s somebody I’d like to know better when the war is over. I find his letters funnier than his books. Tinkerty-tonk, my darling. I pray I’ll hear from you tomorrow. If I don’t I’ll visit your office in a fake beard.

All love, JB

## Checkered Doughnuts

Roll this magic square into a tube by joining the upper and lower edges, then join the ends of the tube. Every row, column, and diagonal on the resulting torus will add to 34.

Bend this chessboard similarly into a torus, then mate in 4.

Hint: The solution comprises only two lines.