Seating Trouble

The Fall 1978 issue of Pi Mu Epsilon Journal included this problem, submitted by Pier Square. Four men are playing bridge. Their names are Banker, Waiter, Baker, and Farmer, and, as it happens, each man’s name is another man’s job. Mr. Baker’s partner is the baker, Mr. Banker’s partner is the farmer, and the waiter sits at Mr. Farmer’s right. Who is sitting at the banker’s left?

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In his 1994 book Mathematics: A Human Endeavor, Harold R. Jacobs gives a similar problem, by French puzzlist Pierre Berloquin. Around a table sit the four members of the street storekeepers’ committee: a grocer, a baker, a tobacconist, and a butcher, whose name is Andre. If Andre sits on Charmeil’s left, Berton sits at the grocer’s right, and Duclos, who faces Charmeil, is not the baker, then what are the professions of the other members? That’s simple enough that it doesn’t require a solution, I think. But here’s a much harder one, composed by J.B. Parker for the January 1941 issue of Eureka: Seven men, Messrs. Black, Blue, Brown, Gray, Green, Purple and White sat down at a circular table laid for eight. Each man was wearing a tie and a pair of socks — and also had a car, their colours being three of the names of the other six men. There was a tie, a pair of socks and a car of each of the seven colours. Mr. Blue sat next to the man with the green tie; between Mr. Gray and the man with white socks there sat a man wearing a white tie, and opposite him sat Mr. Green. Mr. Brown’s socks were of the same colour as the tie of the man who occupied the chair on his right, and Mr. Green wore a brown tie. The empty chair lay between Mr. Black and the man who wore a green tie. The man with black socks had a gray car, and the man with gray socks had a black car. Mr. Purple’s car was the same colour as Mr. Gray’s tie, and this was of the same colour as Mr. White’s socks. The man with the name of the colour of Mr. White’s car wore socks of the colour of Mr. Black’s car, i.e. blue. Mr. Purple’s tie was of the colour of the car of the man who occupied the chair on his right, and Mr. Brown sat opposite the man with the white car. The colour of the socks of the man whose tie was the colour of Mr. Gray’s car was the same as that of the tie of the man whose socks were the colour of Mr. Black’s car, and this colour was not black. Find the colours of the socks, tie, and car of each man. The answer (but not the solution) appears on page 11 of the May 1941 issue. (The Eureka Digital Archive is published under a Creative Commons license.) (Pier Square, “The Bridge Game,” Pi Mu Epsilon Journal 6:9 [Fall 1978], 530.) 05/24/2021 Reader Dharmesh Jain kindly offers this walkthrough of the Eureka puzzle.

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seating trouble solution

In his 1994 book Mathematics: A Human Endeavor, Harold R. Jacobs gives a similar problem, by French puzzlist Pierre Berloquin. Around a table sit the four members of the street storekeepers’ committee: a grocer, a baker, a tobacconist, and a butcher, whose name is Andre. If Andre sits on Charmeil’s left, Berton sits at the grocer’s right, and Duclos, who faces Charmeil, is not the baker, then what are the professions of the other members?

That’s simple enough that it doesn’t require a solution, I think. But here’s a much harder one, composed by J.B. Parker for the January 1941 issue of Eureka:

Seven men, Messrs. Black, Blue, Brown, Gray, Green, Purple and White sat down at a circular table laid for eight. Each man was wearing a tie and a pair of socks — and also had a car, their colours being three of the names of the other six men. There was a tie, a pair of socks and a car of each of the seven colours.

Mr. Blue sat next to the man with the green tie; between Mr. Gray and the man with white socks there sat a man wearing a white tie, and opposite him sat Mr. Green.

Mr. Brown’s socks were of the same colour as the tie of the man who occupied the chair on his right, and Mr. Green wore a brown tie. The empty chair lay between Mr. Black and the man who wore a green tie. The man with black socks had a gray car, and the man with gray socks had a black car.

Mr. Purple’s car was the same colour as Mr. Gray’s tie, and this was of the same colour as Mr. White’s socks. The man with the name of the colour of Mr. White’s car wore socks of the colour of Mr. Black’s car, i.e. blue. Mr. Purple’s tie was of the colour of the car of the man who occupied the chair on his right, and Mr. Brown sat opposite the man with the white car. The colour of the socks of the man whose tie was the colour of Mr. Gray’s car was the same as that of the tie of the man whose socks were the colour of Mr. Black’s car, and this colour was not black.

Find the colours of the socks, tie, and car of each man.

The answer (but not the solution) appears on page 11 of the May 1941 issue. (The Eureka Digital Archive is published under a Creative Commons license.)

(Pier Square, “The Bridge Game,” Pi Mu Epsilon Journal 6:9 [Fall 1978], 530.)

05/24/2021 Reader Dharmesh Jain kindly offers this walkthrough of the Eureka puzzle.

May 1, 2021May 24, 2021 | Puzzles

Unquote

“To convince any man against his will is hard, but to please him against his will is justly pronounced by Dryden to be above the reach of human abilities.” — Samuel Johnson

“Thou canst not joke an Enemy into a Friend; but thou may’st a Friend into an Enemy.” — Ben Franklin

April 30, 2021April 29, 2021 | Quotations · Society

Seeing Red

University of Waterloo mathematician Ross Honsberger chose this problem for his 2004 collection Mathematical Delights; it’s a generalization of a problem that Robert Gebhardt had offered in the Fall 1999 issue of Pi Mu Epsilon Journal. Paint the outside of an n × n × n cube red, then chop it into n³ unit cubes. Put the unit cubes in a box, mix them up thoroughly, withdraw one at random, and throw it across a table. What’s the probability that it comes to rest with a red face on top?

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Credit for the solution goes to Amy Kuiper of Michigan’s Alma College and to the Grand Valley State University Problem Solving Group of Allendale, Mich. The face that receives our final scrutiny might be any one of the faces; all are equally likely. So the probability that it’s red is just $\displaystyle \frac{\textup{the number of red faces}}{\textup{the total number of faces}} = \frac{6n^{2}}{6n^{3}} = \frac{1}{n}.$ (Clayton W. Dodge, “Problem Department,” Pi Mu Epsilon Journal 11:1 [Fall 1999], 47-63. Gebhardt’s is Problem 948, on page 56.)

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April 30, 2021April 29, 2021 | Puzzles

Nowheresville

https://commons.wikimedia.org/wiki/File:Argleton,_view_from_Bold_Lane.JPG — Image: Wikimedia Commons

In September 2008, Mike Nolan, head of web services at Edge Hill University in Ormskirk, England, noticed something strange on Google Maps. “I grew up in the area and spotted on the map one day that it said ‘Argleton’,” he told the Guardian. “But it’s just a farmer’s field close to the village hall and playing fields. I think a footpath goes across the field, but that’s all.”

Bloggers began to discuss the nonexistent town, which found its way into other services that used Google’s data: Employment agencies, weather services, and letting agents began to cite Argleton in their listings, reassigning real people and businesses to the phantom settlement because of its claimed location.

Was it a joke? A placeholder? A misspelling? Whatever it was, it had disappeared again by May 2010. Google would say only that it experiences “occasional errors” and that it gets its mapping information from a Dutch company called Tele Atlas (whose spokesperson would add only, “I really can’t explain why these anomalies get into our database”).

Danny Dorling, president of the Society of Cartographers, said, “I would bet that this is an innocent mistake. In other words, it was not intentionally inserted to catch out anyone infringing the map’s copyright, as some are saying. But the bottom line is that we don’t know what mapping companies do to protect their maps or to hide secret locations, as some are obligated to do.”

April 29, 2021April 28, 2021 | Oddities · Technology

Cutting Cake

mabry proof

In the College Mathematics Journal in 2001, Rick Mabry published this “proof without words” that

$\displaystyle \frac{1}{3} + \frac{1}{3^{2}} + \frac{1}{3^{3}} + \cdots = \frac{1}{2}.$

He gives a charming explanation here.

(Rick Mabry, “Mathematics Without Words,” College Mathematics Journal 32:1 [January 2001], 19.)

April 29, 2021April 28, 2021 | Science & Math

Best Friends

Hence it is that the moral character of a man eminent in letters or in the fine arts is treated, often by contemporaries, almost always by posterity, with extraordinary tenderness. … These friendships are exposed to no danger from the occurrences by which other attachments are weakened or dissolved. … Plato is never sullen. Cervantes is never petulant. Demosthenes never comes unseasonably. Dante never stays too long. No difference of political opinion can alienate Cicero. No heresy can excite the horror of Bossuet. Nothing, then, can be more natural than that a person endowed with sensibility and imagination should entertain a respectful and affectionate feeling towards those great men with whose minds he holds daily communion.

— Thomas Macaulay, An Essay on Francis Bacon, 1837

April 28, 2021April 27, 2021 | Literature

First Things First

It is a traditional axiom of medicine that health is the absence of disease. What is a disease? Anything that is inconsistent with health. If the axiom has any content, a better answer can be given. The most fundamental problem in the philosophy of medicine is, I think, to break the circle with a substantive analysis of either health or disease.

— Christopher Boorse, “Health as a Theoretical Concept,” Philosophy of Science 44:4 (1977), 542-573

April 28, 2021April 27, 2021 | Science & Math

“The Individual and the World”

https://commons.wikimedia.org/wiki/File:EarthRender.png — Image: Wikimedia Commons

There is an eternal antagonism of interest between the individual and the world at large. The individual will not so much care how much he may suffer in this world provided he can live in men’s good thoughts long after he has left it. The world at large does not so much care how much suffering the individual may either endure or cause in this life, provided he will take himself clean away out of men’s thoughts, whether for good or ill, when he has left it.

— Samuel Butler, Notebooks

April 27, 2021April 26, 2021 | Society

The Presidents’ Tree

In August 1865, Maryland farmer Samuel M’Closky Fenton carved this grid into an American beech in Takoma Park:

  	N L O C N I L M L I N C O L N
	L O C N I L M A M L I N C O L
	O C N I L M A H A M L I N C O
	C N I L M A H A H A M L I N C
	N I L M A H A R A H A M L I N
	I L M A H A R B R A H A M L I
	L M A H A R B A B R A H A M L
	I L M A H A R B R A H A M L I
	N I L M A H A R A H A M L I N
	C N I L M A H A H A M L I N C
	O C N I L M A H A M L I N C O
	L O C N I L M A M L I N C O L
	N L O C N I L M L I N C O L N

He intended it as a memorial to Abraham Lincoln, who had been assassinated that April. Starting at the central A and following a jagged path toward any of the corner Ns will spell out the fallen president’s name.

Fenton also carved the name of every American president to date. In 1948 the tree was enclosed in an iron fence and dedicated “as a living memorial to men who gave their lives for their country in the war of 1861-1865.”

April 27, 2021April 26, 2021 | History

In a Word

philobiblist
n. a lover of books

Created by Slovakian artist Matej Kren, the Idiom Installation in Prague’s municipal library seems to present an infinite tower of books, thanks to some conveniently placed mirrors.

It debuted at the São Paulo International Biennial in 1995 and moved permanently Prague in 1998.

April 26, 2021April 25, 2021 | Art · Language · Literature