Room for More

When logician John Venn introduced the famous diagrams that bear his name, he expressed an interest in “symmetrical figures … elegant in themselves.” He thought the “simplest and neatest figure” that showed all possible logical relations among four sets was four equal ellipses arranged like this:

https://commons.wikimedia.org/wiki/File:Venn%27s_four_ellipse_construction.svg — Image: Wikimedia Commons

“It is obvious that we thus get the sixteen compartments that we want, counting, as usual, the outside of them all as one compartment. … The desired condition that these sixteen alternatives shall be mutually exclusive and collectively exhaustive, so as to represent all the component elements yielded by the four terms taken positively and negatively, is of course secured.”

Interestingly, he added that “with five terms combined together ellipses fail us”: Venn believed that it was impossible to create a Venn diagram with five ellipses. Amazingly, that assertion went unchallenged for nearly a century — it was only in 1975 that Branko Grünbaum found a diagram with five ellipses:

https://commons.wikimedia.org/wiki/File:Symmetrical_5-set_Venn_diagram.svg — Image: Wikimedia Commons

It’s not possible to form a Venn diagram with six or more ellipses. Do we even need one with five? According to Reddit, yes, we do:

(Peter Hamburger and Raymond E. Pippert, “Venn Said It Couldn’t Be Done,” Mathematics Magazine 73:2 [April 2000], 105-110.)

September 12, 2018 | Science & Math

Self-Study

For a puzzlers’ party in 1993, University of Wisconsin mathematician Jim Propp devised a “self-referential aptitude test,” a multiple-choice test in which each question except the last refers to the test itself:

1. The first question whose answer is B is question

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

2. The only two consecutive questions with identical answers are questions

(A) 6 and 7
(B) 7 and 8
(C) 8 and 9
(D) 9 and 10
(E) 10 and 11

3. The number of questions with the answer E is

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

The full 20-question test is here, the solution is here, and an interesting collection of solving routes is here.

(Jim Propp, “Self-Referential Aptitude Test,” Math Horizons 12:3 [February 2005], 35.)

September 12, 2018September 11, 2018 | Puzzles

Aplomb

Abe Lincoln’s law partner William Herndon said Lincoln told this joke “often and often”:

Well, there was a party once, not far from here, which was composed of ladies and gentlemen. A fine table was set and the people were greatly enjoying themselves. Among the crowd was one of those men who had audacity — was quick-witted, cheeky, and self-possessed — never off his guard on any occasion. After the men and women had enjoyed themselves by dancing, promenading, flirting, etc., they were told that the table was set. The man of audacity — quick-witted, self-possessed, and equal to all occasions — was put at the head of the table to carve the turkeys, chickens, and pigs. The men and women surrounded the table, and the audacious man, being chosen carver, whetted his great carving knife with the steel and got down to business and commenced carving the turkey, but he expended too much force and let a fart — a loud fart so that all the people heard it distinctly. As a matter of course it shocked all terribly. A deep silence reigned. However, the audacious man was cool and entirely self-possessed; he was curiously and keenly watched by those who knew him well, they suspecting that he would recover in the end and acquit himself with glory. The man, with a kind of sublime audacity, pulled off his coat, rolled up his sleeves, put his coat deliberately on a chair, spat on his hands, took his position at the head of the table, picked up the carving knife and whetted it again, never cracking a smile nor moving a muscle of his face. It now became a wonder in the minds of all the men and women how the fellow was to get out of his dilemma. He squared himself and said loudly and distinctly: ‘Now, by God, I’ll see if I can’t cut up this turkey without farting.’

September 11, 2018 | Humor

Forefathers

A problem from the 1996 mathematical olympiad of the Republic of Moldova:

Twenty children attend a rural elementary school. Every two children have a grandfather in common. Prove that some grandfather has not less than 14 grandchildren in this school.

	Select Click for Answer
Consider the grandfather with the largest number of grandchildren, and suppose, for a contradiction, that that number is less than 14. There’s a child at the school who isn’t his grandchild, and each of his grandchildren shares a grandfather with that child. They can’t all share the same grandfather, because then that man would surpass our “maximal” grandfather. So there’s some third grandfather to divide that duty with him. Now, every student we haven’t spoken about shares a grandfather with one that we have. And that leaves room for no fourth grandfather; every student in the school must count some two of these men as his grandfathers. We have 40 grandfatherings to assign to these 20 students, and 40 = 3 × 13 + 1, so try as we might, we can’t assign 13 or fewer to each of the three men. One of them must have at least 14 grandchildren. (This is all laid out more rigorously at the link below, page 371.) (Olympiad Corner, Crux Mathematicorum 27:6 [October 2001], 357-373.)

Click for Answer

September 11, 2018September 10, 2018 | Puzzles

Noted

https://commons.wikimedia.org/wiki/File:Morrissey_and_Marr_2.jpg — Image: Wikimedia Commons

The French phrase Mort ici, j’en ai marre translates as “Dead here, I’m fed up” and sounds like “Morrissey, Johnny Marr.”

(Thanks, Ian.)

September 10, 2018 | Entertainment · Language

Podcast Episode 216: The Tromelin Island Castaways

https://commons.wikimedia.org/wiki/File:Tromelin_aerial_photograph.JPG — Image: Wikimedia Commons

In 1761 a French schooner was shipwrecked in the Indian Ocean, leaving more than 200 people stranded on a tiny island. The crew departed in a makeshift boat, leaving 60 Malagasy slaves to fend for themselves and wait for rescue. In this week’s episode of the Futility Closet podcast we’ll tell the story of the Tromelin Island castaways, which one observer calls “arguably the most extraordinary story of survival ever documented.”

We’ll also admire some hardworking cats and puzzle over a racer’s death.

See full show notes …

September 10, 2018September 17, 2018 | History · Podcast

The Body Politic

This iconic image of Abraham Lincoln is not authentic — Lincoln never posed in this “heroic” style during his lifetime, so after his death an enterprising artist added Abe’s head to a portrait of South Carolina Democrat John C. Calhoun.

Both men would have been aghast. “Many in the South once believed that slavery was a moral and political evil,” Calhoun once wrote. “That folly and delusion are gone. We see it now in its true light, and regard it as the most safe and stable basis for free institutions in the world.”

September 9, 2018 | History

Magic

A ring that encircles a length of chain will be caught in a loop if it tumbles during its fall. By Newton’s Third Law, when the turning ring strikes the chain it transfers momentum to the loop at the end — which causes it to rise and swallow the ring.

September 8, 2018 | Science & Math

Geometry

“The happiness of London is not to be conceived but by those who have been in it. I will venture to say, there is more learning and science within the circumference of ten miles from where we now sit, than in all the rest of the kingdom.” — Samuel Johnson

“I believe the parallelogram between Oxford Street, Piccadilly, Regent Street and Hyde Park encloses more intelligence and human ability, to say nothing of wealth and beauty, than the world has ever collected in such a space before.” — Sydney Smith

September 7, 2018 | Quotations · Society

Neat

I just ran across this in an old Mathematical Gazette: R.H. Macmillan of Buckinghamshire shared a tidy expression for the area of a triangle whose vertices have coordinates (x₁, y₁), (x₂, y₂), and (x₃, y₃):

$\displaystyle \pm \frac{1}{2}\left \{ x_{1} \left ( y_{2} - y_{3} \right ) + x_{2} \left ( y_{3} - y_{1} \right ) + x_{3} \left ( y_{1} - y_{2} \right ) \right \}$

The sign is positive if the numbering is counterclockwise and negative if it’s clockwise.

“The expression is readily derived geometrically (using only the fact that the sum of the areas on each side of the diagonal of a rectangle must be equal) and so provides an interesting elementary exercise.”

(R.H. Macmillan, “Area of a Triangle,” Mathematical Gazette 77:478 (March 1993), 88.)

September 7, 2018September 6, 2018 | Science & Math