Kriegspiel

Rotenberg kriegspiel problem

Kriegspiel is a variant of chess in which neither player can see the other’s pieces. The two players sit at separate boards, White with the white pieces and Black with the black, and a referee facilitates the game. When a player attempts a move, the referee declares whether it’s legal or illegal. If it’s legal then it stands; if it’s not, the player retracts it and tries again.

This makes for some interesting chess problems. In this example, by Jacques Rotenberg, White knows that there’s a black bishop on a dark square, but he doesn’t know where it is. How can he mate Black in 8 moves?

This is tricky, because if White captures the bishop by accident, the position is stalemate. Accordingly White must avoid bishop or knight moves to begin with. The answer is to try 1. Rg2. If the referee declares that this is illegal, that means that the black bishop is somewhere on the second rank and it’s safe for White to play 1. Nf2, giving mate immediately.

If the referee declares that 1. Rg2 is legal, then the move is made, Black moves his invisible bishop (his king and pawn have no legal moves), and it’s White’s turn again.

Now White announces 2. Rg8. If the referee says that this is illegal, then the black bishop is on the g-file, and White can safely play 2. Be5. Now if Black captures the bishop, then 3. Nf2 is mate; on any other Black move, 3. Nf2+ followed (if necessary) by 4. Rxh2+ is mate.

If 2. Rg8 is legal, then White plays it, Black again inscrutably moves his bishop, and now White plays 3. Rh8. (There’s no danger that he’ll capture the black bishop inadvertently on h8, because it cannot have been on g7 on the previous turn.)

Black moves his invisible bishop again and now White plays 4. Rh5 followed by 5. Rb5 (if that’s not possible then 5. Rh3 and 6. Be5), 6. Rb1, 7. Nf2+ Bxf2 and 8. Kxf2#. White wins in eight moves at most. In order to travel safely from a2 to b1, the white rook must pass through h8!

| Puzzles

A Fool’s Logic

“It is true that you may fool all the people some of the time; you can even fool some of the people all the time; but you can’t fool all of the people all the time.”

This is commonly attributed to Abraham Lincoln, though it’s not clear that he actually said it. In 2004 mathematician Paul Stockmeyer noticed that its meaning is somewhat ambiguous, too. If we use P(x) to denote the predicate “x is a person,” T(y) to denote the predicate “y is a time,” and F(x, y) to denote the two-argument predicate “x is fooled at time y,” then the first phrase of the quotation, “It is true that you may fool all the people some of the time,” might mean either

\displaystyle \forall x\left ( P\left ( x \right ) \Rightarrow \exists y\left ( T\left ( y \right ) \wedge F\left ( x, y \right )\right )\right )

or

\displaystyle \exists y\left ( T\left ( y \right ) \wedge \forall x\left ( P\left ( x \right )\Rightarrow F\left ( x, y \right ) \right )\right ).

The first statement means “For every possible x, if x is a person then there exists a y such that y is a time and moreover x is fooled at time y” (or, more coloquially, “For every person, there is a time when that person is fooled”).

The second means “There exists a y such that y is a time and moreover for every x, if x is a person then x is fooled at time y (or “There is a time when everyone is simultaneously fooled”).

Which is the right interpretation? Stockmeyer polled his classes and found them nearly equally divided. And that’s only the first phrase of the quotation! Does the second phrase, “you can even fool some of the people all the time,” mean that there are people who remain constantly fooled about everything — or that you can always find a fool at any given time?

“However they are interpreted, they serve as a wonderfully effective preparation for his main point contained in the third phrase,” Stockmeyer writes. “And this phrase, with two quantifiers of the same type, is completely unambiguous.”

(Paul K. Stockmeyer, “What Did Lincoln Really Mean?” College Mathematics Journal 35:2 [2004], 103-104.)

| Science & Math

Allestone

https://archive.org/details/fathersmemoirsof01malk/page/n161?q=Allestone

In 1806 British scholar Benjamin Heath Malkin published A Father’s Memoirs of His Child to record the almost alarming gifts of his son Thomas, who had taught himself to read and write by age 2, inquired into mathematics and Latin, and at age 5 invented an imaginary country called Allestone:

Allestone … was so strongly impressed on his own mind, as to enable him to convey an intelligible and lively transcript of its description. Of this delightful territory he considered himself as king. He had formed the project of writing its history, and had executed the plan in detached parts. Neither did his ingenuity stop here; for he drew a map of the country, giving names of his own invention to the principal mountains, rivers, cities, seaports, villages, and trading towns.

“The country is an island,” the father explained, “and therefore the better calculated for the scene of the transactions he has assigned to it. The rivers, for the most part, rise in such situations, and flow in such directions, as they would in reality assume. Their course is marked out with reference to the position of principal towns, and other objects of general convenience.”

Thomas sketched out the country’s political history, principal actors, and monetary system, and had composed a series of representative adventures among its people and a comic opera (“only imaginary music, made by Thomas Williams Malkin, who does not understand real music”), when he died, probably of peritonitis, at age 6 — leaving his subjects without a king.

| Oddities

Trade Secrets

Seems there were three lawyers and three MBAs traveling by train to a conference. At the station, the three MBAs each buy tickets and watch as the three lawyers buy only a single ticket. ‘How are three people going to travel on only one ticket?’ asks an MBA. ‘Watch and you’ll see’ answers a lawyer.

They all board the train. The MBAs take their respective seats but all three lawyers cram into a restroom and close the door behind them. Shortly after the train has departed, the conductor comes around collecting tickets. He knocks on the restroom door and says, ‘Ticket, please.’ The door opens just a crack and a single arm emerges with a ticket in hand. The conductor takes it and moves on.

The MBAs see this and agree it was quite a clever idea. So after the conference, the MBAs decide to copy the lawyers on the return trip and save some money (being clever with money and all that). When they get to the station, they buy a single ticket for the return trip. To their astonishment, the lawyers don’t buy a ticket at all. ‘How are you going to travel without a ticket?’ asks one perplexed MBA. ‘This time we can’t tell you,’ says one of the lawyers, ‘it’s a professional secret.’

When they all board the train the three MBAs cram into a restroom and the three lawyers cram into another one nearby. The train departs. Shortly afterward, one of the lawyers leaves his restroom and walks over to the restroom where the MBAs are hiding. He knocks on the door and says, ‘Ticket please.’

— Marc Galanter, Lowering the Bar: Lawyer Jokes and Legal Culture, 2005

| Humor

Relative

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Navigators from the Poluwat atoll of Micronesia find their way among islands by appealing to parallax — a reference island at one side of their course will appear to pass beneath a succession of stars:

The star bearings of the reference island from both the starting and ending points of the trip are known, since on another occasion the reference island may itself become a destination. In between there are other navigation star positions under which the reference island will pass as it ‘moves’ backwards. Its passage under each of these stars marks the end of one etak and the beginning of another. Thus the number of star positions which lie between the bearing of the reference island as seen from the island of origin and its bearing as seen from the island of destination determine the number of etak, which can here be called segments, into which the voyage is conceptually divided. When the navigator envisions in his mind’s eye that the reference island is passing under a particular star he notes that a certain number of segments have completed and a certain proportion of the voyage has therefore been accomplished.

This is a dynamic model: Where Western navigators think of a vessel moving among stationary islands, the Poluwatese find it more natural to think of the canoe as stationary and the islands as moving around it. “Etak is perfectly adapted for its use by navigators who have no instruments, charts, or even a dry place in which to spread a chart if they had one,” writes Stephen D. Thomas in The Last Navigator. “Etak allows the Micronesian navigator to process all his information — course, speed, current drift, and so on — through a single, sea-level perspective.”

(Thomas Gladwin, East Is a Big Bird: Navigation and Logic on Puluwat Atoll, 1970, quoted in Lorenzo Magnani, Philosophy and Geometry, 2001.)

| Science & Math

The Miser and His Gold

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There was a miser who sold his property and bought a lump of gold. The man then buried his gold just outside the city walls, where he constantly went to visit and inspect it. One of the workmen noticed the man’s behaviour and suspected the truth. Accordingly, after the man had gone away, he took the gold. When the man came back and found that the hiding-place was empty, he began to cry and tear his hair. Someone saw the man’s extravagant grief and asked him what was wrong. Then he said to the man, “Enough of your grieving! Take a stone and put it where the gold was, and make believe the gold is still there: it’s not as if you ever made any use of it!”

— Aesop

| Literature · Society

A Modest Proposal

https://www.flickr.com/photos/home_of_chaos/7609870922
Image: Flickr

While a visiting fellow at All Souls College, Oxford, in 1978, Claude Shannon pondered a personal challenge he faced there:

An American driving in England is confronted with a wild and dangerous world. The cars have the driver on the right and he is supposed to drive on the left side of the road. It is as though English driving is a left-handed version of the right-handed American system.

I can personally attest to the seriousness of this problem. Recently my wife and I, together with another couple on an extended visit to England, decided to jointly rent a car. … With our long-ingrained driving habits the world seemed totally mad. Cars, bicycles and pedestrians would dart out from nowhere and we would always be looking in the wrong direction. The car was usually filled with curses from the men and with screams and hysterical laughter from the women as we careened from one narrow escape to another.

His solution was “grandiose and utterly impractical — the idle dream of a mathematician”:

How will we do this? In a word, with mirrors. If you hold your right hand in front of a mirror, the image appears as a left hand. If you view it in a second mirror, after two reflections it appears now as a right hand, and after three reflections again as a left hand, and so on.

Our general plan is to encompass our American driver with mirror systems which reflect his view of England an odd number of times. Thus he sees the world about him not as it is but as it would be after a l80° fourth-dimensional rotation.

A corresponding adjustment to the steering system will turn the car left when the driver steers right, and vice versa. And filling the cabin with a high-density liquid will reverse the feeling of centrifugal force as well. “A snorkel provides for his breathing and altogether, with our various devices, he feels very much as though he were at home in America!”

(Claude E. Shannon, “The Fourth-Dimensional Twist, or a Modest Proposal in Aid of the American Driver in England,” typescript, All Souls College, Oxford, Trinity term, 1978; via Jimmy Soni and Rob Goodman, A Mind at Play: How Claude Shannon Invented the Information Age, 2017.)

| Science & Math

Podcast Episode 221: The Mystery Man of Essex County

https://commons.wikimedia.org/wiki/File:Debosnys-Self-Portrait.png

In 1882, a mysterious man using a false name married and murdered a well-to-do widow in Essex County, New York. While awaiting the gallows he composed poems, an autobiography, and six enigmatic cryptograms that have never been solved. In this week’s episode of the Futility Closet podcast we’ll examine the strange case of Henry Debosnys, whose true identity remains a mystery.

We’ll also consider children’s food choices and puzzle over a surprising footrace.

See full show notes …

| Crime · Podcast · Puzzles

A Point of Law

In 1864, two Englishmen entered into an agreement: Raffles would procure 125 bales of fine cotton from India and deliver them to Wichelhaus, who would buy them for a fixed price. They agreed that the cotton would arrive aboard the ship Peerless.

By a sublime coincidence, there were two ships named Peerless sailing from Bombay to Liverpool that year. Wichelhaus had in mind the one that set sail in October, where Raffles had intended another one in December. When his cotton arrived two months later than he’d expected, Wichelhaus refused to accept it, and Raffles sued him.

Who’s right? Raffles had delivered the cotton in good faith according to their written agreement, but Wichelhaus argued that he was entitled to his own understanding of an ambiguous term. Raffles hadn’t met that, so Wichelhaus wasn’t obligated to pay him.

In the end Wichelhaus prevailed: The court ruled that because of the overlooked ambiguity the two men had not had the same transaction in mind when they’d made their agreement — so there was no binding contract.

(Raffles v. Wichelhaus, 2 Hurl. & C. 906 [1864].)

| Society