## 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.

## Disqualified

From Clark Kinnaird’s *Encyclopedia of Puzzles and Pastimes*, 1946:

Though a great American, Wendell Willkie nevertheless lacked one of the four necessary requirements for becoming President of the United States. One must be at least 35, a native-born American, and a resident of the U.S.A. for at least 14 years. Name the fourth requirement which Willkie also lacked?

## Roundup

Two old chestnuts:

1. I have six pieces of chain, each consisting of four links. It costs 10 cents to cut open one link and 25 cents to weld it together again. What will it cost to have the six pieces joined into one chain?

2. The two volumes of my scintillating autobiography stand side by side in order on a shelf. A bookworm starts at page 1 of volume 1 and eats his way in a straight line to the last page of volume 2. If each cover is 1/8 of an inch thick, and each book without the covers is 2 inches thick, how far does the bookworm travel?

And two trick questions:

1. What is the name of the title character in *The Merchant of Venice*?

2. Who played Frankenstein in the 1931 American film?

## Thin Thinking

Some of the figures (particularly the holy ones) in El Greco paintings seem unnaturally tall and thin. An ophthalmologist surmised that the painter had a defect of vision that caused him to see people this way.

The zoologist Sir Peter Medawar pointed out that we can reject this conjecture on purely logical grounds. What was his insight?

## On Target

Samuel Isaac Jones offered this poser in his 1929 book *Mathematical Wrinkles*:

Cook was within 10 miles of the north pole and Peary was also within 10 miles of the pole, but 20 miles from Cook. What direction was Peary from Cook? Suppose Peary threw a ball at Cook and hit him. In what direction did the ball go?

He omitted the answer, apparently inadvertently. What is it?

## A Missed Opportunity

By Bruno Sommer, 1910. White has just moved, and he realizes too late that he could have mated Black on the move. What was his last move, and what was the mate?

## The Crossing

A family of four has to cross a river. The father and mother each weigh 150 pounds, and each of the two sons weighs 75 pounds. Unfortunately, the boat will carry only 150 pounds maximum. How can they get across?

## “The Best Bridge Problem Ever Invented”

I don’t play bridge, so I’m posting this somewhat blindly. It was devised by W.H. Whitfeld, card editor of the *Field*, apparently in the late 19th century. The reader who submitted it to the *Strand* wrote, “If you don’t know the solution, I guarantee that it will take you or any of your staff three or four days.”

“We have a higher opinion of our readers’ skill than to allot them such a time-limit as this,” wrote the editors. “But certainly anyone who can solve this problem in three or four *hours* will have good cause to be congratulated on his ingenuity.”

## A French Cryptogram

When the Chevalier de Rohan was sent to the Bastille in 1674 on suspicion of treason, he knew there was no evidence against him except what might be extracted from one other prisoner. His friends had promised to communicate the result of that examination, and in sending him some fresh clothing they wrote on one of the shirts MG DULHXCCLGU GHJ YXUJ, LM CT ULGC ALJ.

For 24 hours de Rohan puzzled over the message, but he could make no sense of it. Despairing, he admitted his guilt and was executed. What was the message?

## “The Hidden Star”

From Henry Dudeney:

The ilustration represents a square tablecloth of choice silk patchwork. This was put together by the members of a family as a little birthday present for one of its number. One of the contributors supplied a portion in the form of a perfectly symmetrical star, and this has been worked in exactly as it was received. But the triangular pieces so confuse the eye that it is quite a puzzle to find the hidden star.

Can you discover it, so that, if you wished, by merely picking out the stitches, you could extract it from the other portions of the patchwork?