## Half and Half

Choose an arbitrary point inside an equilateral triangle and draw segments to the vertices and perpendiculars to the sides. This divides the triangle into six smaller triangles, A, B, C, D, E, and F. Prove that the areas A + C + E and B + D + F are equal.

## Hostilities

A 10×10 chessboard contains 41 rooks. Prove that there are five rooks that don’t attack one another.

## Square Deal

A “coffin,” or killer problem, from the oral entrance exams to the math department of Moscow State University:

Construct (with ruler and compass) a square given one point from each side.

## Black and White

By A.B. Arnold, from the 1859 problem tournament of the *New York Clipper*. White to mate in two moves.

## Pigeon Puzzle

In 1982, 74-year-old David Martin found the skeleton of a carrier pigeon in the chimney of his house in Bletchingley, Surrey. Attached to its leg was an encrypted message believed to have been sent from France on D-Day, June 6, 1944:

AOAKN HVPKD FNFJW YIDDC

RQXSR DJHFP GOVFN MIAPX

PABUZ WYYNP CMPNW HJRZH

NLXKG MEMKK ONOIB AKEEQ

WAOTA RBQRH DJOFM TPZEH

LKXGH RGGHT JRZCQ FNKTQ

KLDTS FQIRW AOAKN 27 1525/6

What does it mean? No one knows — the message still hasn’t been deciphered.

“Although it is disappointing that we cannot yet read the message brought back by a brave carrier pigeon,” announced Britain’s Government Communications Headquarters last November, “it is a tribute to the skills of the wartime code makers that, despite working under severe pressure, they devised a code that was undecipherable both then and now.”

UPDATE: Gord Young of Peterborough, Ontario, claimed to have cracked the code last month using a World War I code book that he had inherited from his great-uncle. He believes the report was written by 27-year-old Lancashire Fusilier William Stott, who had been dropped into Normandy to report on German positions. Stott was killed a few weeks after the report. Here’s Young’s solution:

AOAKN – Artillery Observer At “K” Sector, Normandy

HVPKD – Have Panzers Know Directions

FNFJW – Final Note [confirming] Found Jerry’s Whereabouts

DJHFP – Determined Jerry’s Headquarters Front Posts

CMPNW – Counter Measures [against] Panzers Not Working

PABLIZ – Panzer Attack – Blitz

KLDTS – Know [where] Local Dispatch Station

27 / 1526 / 6 – June 27th, 1526 hours

Young say that the portions that remain undeciphered may have been inserted deliberately in order to confuse Germans who intercepted the message. “We stand by our statement of 22 November 2012 that without access to the relevant codebooks and details of any additional encryption used, the message will remain impossible to decrypt,” a GCHQ spokesman told the BBC on Dec. 16. But he said they would be happy to look at Young’s proposed solution.

(Thanks, John and Ivan.)

## The Chelsea Pensioners

Another puzzle by Lewis Carroll:

In a group of soldiers, if 70 percent have lost an eye, 75 percent an ear, 80 percent an arm, 85 percent a leg, what percentage, *at least*, must have lost all four?

## Project Management

A puzzle by Polish mathematician Paul Vaderlind:

If a blacksmith requires five minutes to put on a horseshoe, can eight blacksmiths shoe 10 horses in less than half an hour? The catch: A horse can stand on three legs, but not on two.

## Swine Wave

A bewildering puzzle by Lewis Carroll: Place 24 pigs in these sties so that, no matter how many times one circles the sties, he always find that the number in each sty is closer to 10 than the number in the previous one.

## Belt Loop

A puzzle by Harry Langman:

A thin belt is stretched around three pulleys, each of which is 2 feet in diameter. The distances between the centers of the pulleys are 6 feet, 9 feet, and 13 feet. How long is the belt?

## No Answer

Helen Fouché Gaines’ 1956 textbook *Cryptanalysis: A Study of Ciphers and Their Solution* concludes with a cipher that, she says, “nobody has ever been able to decrypt”:

VQBUP PVSPG GFPNU EDOKD XHEWT IYCLK XRZAP VUFSA WEMUX GPNIV QJMNJ JNIZY KBPNF RRHTB WWNUQ JAJGJ FHADQ LQMFL XRGGW UGWVZ GKFBC MPXKE KQCQQ LBODO QJVEL.

It was still unsolved in 1968, when Dmitri Borgmann, editor of the *Journal of Recreational Linguistics*, urged his readers to tackle the problem: “Are you going to let this challenge lie there, taunting you for the rest of your lives? Or are you going to get busy and solve that pesky little crypt?”

So far as I can tell, they let it lie there, and it remains unsolved to this day. Any ideas? There are few clues in Gaines’ book. The cipher is the last in a series of exercises at the end of a chapter titled “Investigating the Unknown Cipher,” and she gives no hint as to its source. Of the exercises, she writes, “There is none in which the system may not be learned through analysis, unless perhaps the final unnumbered cryptogram.” The solution says simply “Unsolved.”

## Pillow Problem

From a 1921 essay by A.A. Milne:

TERALBAY is not a word which one uses much in ordinary life. Rearrange the letters, however, and it becomes such a word. A friend — no, I can call him a friend no longer — a person gave me this collection of letters as I was going to bed and challenged me to make a proper word of it. He added that Lord Melbourne — this, he alleged, is a well-known historical fact — Lord Melbourne had given this word to Queen Victoria once, and it had kept her awake the whole night. After this, one could not be so disloyal as to solve it at once. For two hours or so, therefore, I merely toyed with it. Whenever I seemed to be getting warm I hurriedly thought of something else. This quixotic loyalty has been the undoing of me; my chances of a solution have slipped by, and I am beginning to fear that they will never return. While this is the case, the only word I can write about is TERALBAY.

The answer is not RATEABLY, or BAT-EARLY, which “ought to mean something, but it doesn’t.” Rudolf Flesch notes that TRAYABLE is not a word, and that, though TEARABLY appears in small type in *Webster’s Unabridged*, “it obviously won’t do.”

What’s the answer? There’s no trick — it’s an ordinary English word.

## Hole Hearted

Martin Gardner called this the proudest puzzle of his own devising. When the pieces on the left are rearranged as on the right, a hole appears in the center of the square. How is this possible?

“I haven’t the foggiest notion of how to succeed in inventing a good puzzle,” he told the *College Mathematics Journal*. “I don’t think psychologists understand much either about how mathematical discoveries are made. … The creative act is still a mystery.”

## Set Theory

A puzzle by Polish mathematician Paul Vaderlind:

Andre Agassi and Boris Becker are playing tennis. Agassi wins the first set 6-3. If there were 5 service breaks in the set, did Becker serve the first game?

(Service changes with each new game in the 9-game set. A service break is a game won by the non-server.)

## Black and White

By Werner Keym, from *Die Schwalbe*, 1979. What were the last moves by White and Black?