I just ran across this, offered by Morton C. Schwartz in an old issue of Pi Mu Episilon Journal:
Take any number of zeros and any number of ones and place them in a circle, in any order. Reproduce the circle a second time, concentrically with the first. Rotate either circle, and any number of places. The number of zeros opposite ones will always be even.
(Morton C. Schwartz, “An Amazing Parity Theorem,” Pi Mu Episilon Journal 5:7 [Fall 1972], 338.)
https://www.youtube.com/watch?v=KcidhLw94_k
Cape Town artist Philip Barlow depicts “out-of-focus” cityscapes and beach scenes in oil paints. In photography this effect is known as bokeh, the blurring of elements that lie outside the depth of field. Others have said that Barlow renders the world as it might appear to a near-sighted person.
In this sense the subjects of his paintings are not the objects they depict but rather light itself. “The figures in the landscape serve as carriers and reflectors of the light that falls upon them. Bathed in the luminosity, it is my hope that they would become more beautiful. To me, light is the ultimate subject because it embodies the pinnacle of all reality.”
In James Joyce’s Ulysses, as Bloom attends Dignam’s funeral, an odd thought passes through his mind: “Now who is that lankylooking galoot over there in the macintosh? Now who is he I’d like to know? Now, I’d give a trifle to know who he is. Always someone turns up you never dreamt of.” The interloper’s presence seems significant: “Mr Bloom stood far back, his hat in his hand, counting the bared heads. Twelve. I’m thirteen. No. The chap in the macintosh is thirteen. Death’s number. Where the deuce did he pop out of? He wasn’t in the chapel, that I’ll swear. Silly superstition that about thirteen.”
He turns up again later: “In Lower Mount street a pedestrian in a brown macintosh, eating dry bread, passed swiftly and unscathed across the viceroy’s path.”
And still later: “A man in a brown macintosh springs up through a trapdoor.”
Altogether the mysterious man is mentioned 11 times in the novel. In the Cyclops episode we’re told, “The man in the brown macintosh loves a lady who is dead,” and in Ithaca, a catechism of questions and answers, we’re asked, “What selfinvolved enigma did Bloom risen, going, gathering multicoloured multiform multitudinous garments, voluntarily apprehending, not comprehend? Who was M’Intosh?”
The question has never been answered definitively. But in his Cornell University lectures on Ulysses, Vladimir Nabokov suggested that the “chap in the macintosh” is none other than James Joyce himself. In the library episode, “Scylla and Charybdis,” Stephen Dedalus explains that Shakespeare “has hidden his own name, a fair name, William, in the plays, a super here, a clown there, as a painter of old Italy set his face in a dark corner of his canvas.” This “is exactly what Joyce has done — setting his face in a dark corner of his canvas. The Man in the Brown Macintosh who passes through the dream of the book is no other than the author himself. Bloom glimpses his maker!”
Churchill himself is a talented heckler. Sir William Joynson-Hicks was making a speech before Commons and noticed Churchill shaking his head so vigorously that attention was distracted from the address. ‘I see my right honourable friend shaking his head,’ cried Joynson-Hicks with exasperation. ‘I wish to remind him that I am only expressing my own opinion!’
‘And I wish to remind the speaker that I am only shaking my own head,’ replied Churchill.
— Brisbane Courier-Mail, 1952
During the last visit which [Samuel Johnson] made to Lichfield [in 1781], the friends with whom he was staying missed him one morning at the breakfast-table. On inquiring after him of the servants, they understood he had set off from Lichfield at a very early hour, without mentioning to any of the family whither he was going. The day passed without the return of the illustrious guest, and the party began to be very uneasy on his account, when, just before the supper-hour, the door opened, and the Doctor stalked into the room. A solemn silence of a few minutes ensued, nobody daring to inquire the cause of his absence, which was at length relieved by Johnson addressing the lady of the house in the following manner: ‘Madam, I beg your pardon for the abruptness of my departure from your house this morning, but I was constrained to it by my conscience. Fifty years ago, madam, on this day, I committed a breach of filial piety, which has ever since lain heavy on my mind, and has not till this day been expiated. My father, you recollect, was a bookseller, and had long been in the habit of attending market, and opening a stall for the sale of his books during that day. Confined to his bed by indisposition, he requested me, this time fifty years ago, to visit the market, and attend the stall in his place. But, madam, my pride prevented me from doing my duty, and I gave my father a refusal. To do away the sin of this disobedience, I this day went in a post-chaise to Uttoxeter, and going into the market at the time of high business, uncovered my head, and stood with it bare an hour before the stall which my father had formerly used, exposed to the sneers of the standers-by and the inclemency of the weather; a penance by which I trust I have propitiated Heaven for this only instance, I believe, of contumacy toward my father.’
— Richard Warner, A Tour Through the Northern Counties of England, 1802
The act is commemorated today in the Johnson Memorial, which stands in the Market Place, in the town center.
To create his 1970 novel A Humument, British artist Tom Phillips began with W.H. Mallock’s forgotten 1892 novel A Human Document and drew, painted, and collaged over the pages, leaving a few words showing to tell a new, hitherto unrevealed story. For instance, the title arises from Phillips’ deletion of two central syllables in Mallock’s title, and the protagonist, Bill Toge, can appear only when the word “together” or “altogether” arises in Mallock’s original text.
Even this new text is evolving. Phillips has published five editions of the book, in each of which he replaces certain pages; eventually he hopes to replace every page, creating an entirely new work (or an entirely new version of the same work).
“Can we call what Phillips is doing ‘writing’, or would some other term be better?” asked Adam Smyth in the London Review of Books. “What version of authorship or creativity is at work here? A Humument is a reminder that books are inevitably intertextual — they grow out of older texts — and that all writing involves selecting words from a finite pool: what appears to be a constraint, having to work within the walls of an existing novel, in fact dramatises a condition of literature.”
The full text of the 1970 edition is here.
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:
“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:
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.)
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.)
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.’
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.
