Ships That Pass

For four months in 1840 Edgar Allan Poe conducted a puzzle column for the Philadelphia newspaper Alexander’s Daily Messenger. In that time he defied his readers to send him a cryptogram that he could not solve, and at the end of his tenure he declared himself undefeated. One of the later challenges came from 17-year-old Schuyler Colfax of New Carlisle, Iowa, who would grow up to become vice president of the United States:

Dear Sir — As you have in your Weekly Messenger defied the world to puzzle you by substituting arbitrary signs, figures, etc. for the different letters of the alphabet, I have resolved to try my utmost to corner you and your system together, and have manufactured the two odd looking subjects which accompany this as avant couriers. … If you succeed in solving the accompanying, I will, of course, as you request, acknowledge it publicly to my friends.

Poe responded: “We have only time, this week, to look at the first and longest cypher — the unriddling of which, however, will no doubt fully satisfy Mr. Colfax that we have not been playing possum with our readers.” Here’s Colfax’s cryptogram:

8n()58†d w!0 b† !x6n†z k65 !nz k65,8l†n b)x 8nd)Pxd !zw8x 6k n6 36w-†nd!x86n;

x=†0 z†,5!z† x=† w8nz 8n 8xd 62n †dx††w !nz k653† 8x x6 5†36l†5 8xd P†l†P b0 5†l†n,†.


What’s the solution?

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Warming Up,_from_A_Memoir_of_Jane_Austen_(1870).jpg

Chapter 1 of Jack and Alice, a novel written by Jane Austen when she was 13 years old:

Mr. Johnson was once upon a time about 53; in a twelvemonth afterwards he was 54, which so much delighted him that he was determined to celebrate his next Birth day by giving a Masquerade to his Children and Freinds. Accordingly on the Day he attained his 55th year tickets were dispatched to all his Neighbours to that purpose. His acquaintance indeed in that part of the World were not very numerous as they consisted only of Lady Williams, Mr and Mrs Jones, Charles Adams and the 3 Miss Simpsons, who composed the neighbourhood of Pammydiddle and formed the Masquerade.

Before I proceed to give an account of the Evening, it will be proper to describe to my reader, the persons and Characters of the party introduced to his acquaintance.

Mr and Mrs Jones were both rather tall and very passionate, but were in other respects, good tempered, wellbehaved People. Charles Adams was an amiable, accomplished and bewitching young Man; of so dazzling a Beauty that none but Eagles could look him in the Face.

Miss Simpson was pleasing in her person, in ther Manners and in her Disposition; an unbounded ambition was her only fault. Her second sister Sukey was Envious, Spitefull and Malicious. Her person was short, fat and disagreeable. Cecilia (the youngest) was perfectly handsome but too affected to be pleasing.

In Lady Williams every virtue met. She was a widow with a handsome Jointure and the remains of a very handsome face. Tho’ Benevolent and Candid, she was Generous and sincere; Tho’ Pious and Good, she was Religious and amiable, and Tho’ Elegant and Agreable, she was Polished and Entertaining.

The Johnsons were a family of Love, and though a little addicted to the Bottle and the Dice, had many good Qualities.

Such was the party assembled in the elegant Drawing Room of Johnson Court, amongst which the pleasing figure of a Sultana was the most remarkable of the female Masks. Of the Males a Mask representing the Sun, was the most universally admired. The Beams that darted from his Eyes were like those of that glorious Luminary tho’ infinitely Superior. So strong were they that no one dared venture within half a mile of them; he had therefore the best part of the Room to himself, its size not amounting to more than 3 quarters of a mile in length & half a one in breadth. The Gentleman at last finding the feirceness of his beams to be very inconvenient to the concourse by obliging them to croud together in one corner of the room, half shut his eyes by which means, the Company discovered him to be Charles Adams in his plain green Coat, without any mask at all.

When their astonishment was a little subsided their attention was attracted by 2 Domino’s who advanced in a horrible Passion; they were both very tall, but seemed in other respects to have many good qualities. ‘These said the witty Charles, these are Mr and Mrs Jones.’ and so indeed they were.

No one could imagine who was the Sultana! Till at length on her addressing a beautifull Flora who was reclining in a studied attitude on a couch, with ‘Oh Cecilia, I wish I was really what I pretend to be’, she was discovered by the never failing genius of Charles Adams, to be the elegant but ambitious Caroline Simpson, and the person to whom she addressed herself, he rightly imagined to be her lovely but affected sister Cecilia.

The Company now advanced to a Gaming Table where sat 3 Dominos (each with a bottle in their hand) deeply engaged, but a female in the character of Virtue fled with hasty footsteps from the shocking scene, whilst a little fat woman representing Envy, sate alternately on the foreheads of the 3 Gamesters. Charles Adams was still as bright as ever; he soon discovered the party at play to be the 3 Johnsons, Envy to be Sukey Simpson and Virtue to be Lady Williams.

The Masks were then all removed and the Company retired to another room, to partake of elegant and well managed Entertainment, after which the bottle being pretty briskly pushed about by the 3 Johnsons, the whole party not excepting even Virtue were carried home, Dead Drunk.

“Jane Austen was born before those bonds which (we are told) protected women from truth, were burst by the Brontës or elaborately untied by George Eliot,” wrote G.K. Chesterton. “Yet the fact remains that Jane Austen knew more about men than either of them. Jane Austen may have been protected from truth: but it was precious little of truth that was protected from her.”

For the Record

At one point in Samuel Beckett’s 1951 novel Molloy, the title character finds himself at the seaside and “lays in a store of sucking stones”:

They were pebbles but I call them stones. Yes, on this occasion I laid in a considerable store. I distributed them equally between my four pockets, and sucked them turn and turn about. This raised a problem which I first solved in the following way. I had say sixteen stones, four in each of my four pockets these being the two pockets of my trousers and the two pockets of my greatcoat. Taking a stone from the right pocket of my greatcoat, and putting it in my mouth, I replaced it in the right pocket of my greatcoat by a stone from the right pocket of my trousers, which I replaced by a stone from the left pocket of my trousers, which I replaced by a stone from the left pocket of my greatcoat, which I replaced by the stone which was in my mouth, as soon as I had finished sucking it. Thus there were still four stones in each of my four pockets, but not quite the same stones.

It occurs to him that this method won’t ensure that every stone is eventually sucked, and he works out a plan that will achieve this. This takes eight pages, “one of the longest and most detailed accounts of someone working at a mathematical problem in a work of fiction,” according to Richard Phillips in Numbers: Facts, Figures and Fiction.

Maddeningly, in the end Molloy throws away all the stones but one, “for they all tasted exactly the same.”

Banishing Gloom

quin historical atlas

For his Historical Atlas of 1830, Edward Quin took a different approach than other cartographers: Rather than present history as a series of discrete moments, he illustrates the growth of knowledge by covering the earth in obscuring clouds that are beaten back from panel to panel.

“In Quin’s Historical Atlas, the world is shown first in darkness, with clouds obscuring everything outside the Garden of Eden,” note Anthony Grafton and Daniel Rosenberg in Cartographies of Time. “Gradually, as history reveals more of the world, the clouds roll back. Turning the pages of the atlas is a bit like riffling through a flip book, watching darkness recede and the world known to Europeans grow.”

“A Cheap Correspondence”

A curious story of two poor lovers, whose system of correspondence was confined to an ingenious cipher of ink-blots on the outside of the letter, is told by the Poet Coleridge. In one of his walks in the Lake district, he saw the postman offer a letter to the servant-girl at a village inn, who, after carefully looking at the address, returned the document to the postman, telling him that she could not take it in, as she was too poor to pay the postage. Thereupon, Coleridge stepped forward, and giving the postman the shilling required for the letter, handed it to the girl. To his surprise, she did not appear as pleased as he had expected; and when the postman was out of hearing, she explained the matter by confessing to the poet that the whole of the letter consisted in its address and certain exterior blots and marks, and that it was the method adopted by her lover and herself to keep up an unpaid-for correspondence in the days of dear postage.

— Thomas Hood, The Book of Modern English Anecdotes, 1872

Little Wars

H.G. Wells played war games. In 1913 he published a set of rules for playing with miniature infantry, cavalry, and artillery, worked out with his friend Jerome K. Jerome while playing with toy soldiers after lunch one day.

He fired that day a shot that still echoes round the world. An affair — let us parallel the Cannonade of Valmy and call it the Cannonade of Sandgate — occurred, a shooting between opposed ranks of soldiers, a shooting not very different in spirit — but how different in results! — from the prehistoric warfare of catapult and garter. ‘But suppose,’ said his antagonists; ‘suppose somehow one could move the men!’ and therewith opened a new world of belligerence.

With another friend and a lot of playtesting, he worked out a set of rules by which two players contend for control of a battlefield, “little brisk fights in which one may suppose that all the ammunition and food needed are carried by the men themselves.”

In two or three moves the guns are flickering into action, a cavalry melee may be in progress, the plans of the attack are more or less apparent, here are men pouring out from the shelter of a wood to secure some point of vantage, and here are troops massing among farm buildings for a vigorous attack. The combat grows hot round some vital point. Move follows move in swift succession. One realises with a sickening sense of error that one is outnumbered and hard pressed here and uselessly cut off there, that one’s guns are ill-placed, that one’s wings are spread too widely, and that help can come only over some deadly zone of fire.

When Wells published his rules in August 1913, the Spectator raved that “there can be no doubt at all as to the excellence of Little Wars as a game for its own sake” — “Mr. Wells describes his new game and sets out its rules so attractively, and has, moreover, added to his description such alluring photographs, that his readers will find it hard indeed not to hurry out to the toy-shop round the corner, raise the necessary levies, and fall down forthwith upon hands and knees to emulate his achievements in the Battle of Hook’s Farm.”

Happily, since it was published a century ago, the whole thing is in the public domain — it’s available at Project Gutenberg.

Through the Looking-Glass

In 2015, to celebrate the 150th anniversary of the publication of Alice’s Adventures in Wonderland, master sculptor Karen Mortillaro created 12 new sculptures, one for each chapter in Lewis Carroll’s masterpiece. Each takes the form of a table topped with an S-cylindrical mirror, with a bronze sculpture on either side. The sculpture that stands before the mirror is anamorphic, so that the curved mirror’s reflection “undistorts” it, giving it meaning:

“The S-cylindrical mirror is ideal for this project because it allows for the figures on one side of the mirror to be sculpted realistically, while those on the opposite side of the mirror are distorted and unrecognizable,” Mortillaro writes. “The mirror is symbolic of the parallel worlds that Alice might have experienced in her dream state; the world of reality is on one side of the mirror; and the world of illusion is on the mirror’s opposite side.”

Mortillaro’s article appears in the September 2015 issue of Recreational Mathematics Magazine.

Split Decision

In December 2013 a U.S. District Court decided that copyright in the fictional characters Sherlock Holmes and Dr. Watson had expired, but only for the characters as they’re depicted in the earlier novels by Arthur Conan Doyle. Aspects of the characters that are mentioned only in the later novels — such as Dr. Watson’s athletic background, first described in a 1924 short story — are considered new “increments of expression” of those characters, and remain protected.

That makes eminent sense for writers and lawyers, but what about poor Dr. Watson, anxiously stirring the fire at 221B Baker Street? Does he have an athletic background or doesn’t he? The copyright law seems to apply to a version of him that does, and not to one that doesn’t. Should we say there are two Dr. Watsons? That doesn’t seem right.

Worse, “If an author now wants to write a new Holmes novel, but is prohibited from mentioning almost everything pertaining to Professor Moriarty (who only rose to prominence in the later work Valley of Fear), how can we say that he is still writing about the ‘the same’ Holmes, given how much his character was formed through the interaction with his nemesis?” ask legal scholars Burkhard Schafer and Jane Cornwell. “Does this not render any new Holmes necessarily ‘incomplete,’ that is lacking character traits and memories Holmes is ‘known to’ possess, according to the canonical work?”

Even the “public domain” Holmes seems to multiply in this light. We learn that Holmes has an older brother, Mycroft, in “The Adventure of the Greek Interpreter,” published in 1893. But if Mycroft is older than Sherlock, then surely he’s been Sherlock’s brother ever since Sherlock’s birth in 1854. Does the early Sherlock (in, say, A Study in Scarlet) have a brother?

(Burkhard Schafer and Jane Cornwell, “Law’s Fictions, Legal Fictions and Copyright Law,” in Maksymilian Del Mar and William Twining, eds., Legal Fictions in Theory and Practice, 2015.)


A harmonic progression is a progression formed by taking the reciprocals of an arithmetic progression (so an example is 1/1, 1/2, 1/3, 1/4 …). When tutoring mathematics at Oxford, Charles Dodgson had a favorite example to illustrate this:

According to him, it is (or was) the rule at Christ Church that, if an undergraduate is absent for a night during term-time without leave, he is for the first offence sent down for a term; if he commits the offence a second time, he is sent down for two terms; if a third time, Christ Church knows him no more. This last calamity Dodgson designated as ‘infinite.’ Here, then, the three degrees of punishment may be reckoned as 1, 2, infinity. These three figures represent three terms in an ascending series of Harmonic Progression, being the counterparts of 1, 1/2, 0, which are three terms in a descending Arithmetical Progression.

— Lionel A. Tollemache, “Reminiscences of ‘Lewis Carroll,'” Literature, Feb. 5, 1898

Image: Wikimedia Commons

In 1965, as they were writing the first draft of 2001: A Space Odyssey, Stanley Kubrick showed Arthur C. Clarke a set of 12 plastic tiles. Each tile consisted of five squares joined along their edges. These are known as pentominoes, and a set of 12 includes every possible such configuration, if rotations and reflections aren’t considered distinct. The challenge, Kubrick explained, is to fit the 12 tiles together into a tidy rectangle. Because 12 five-square tiles cover 60 squares altogether, there are four possible rectangular solutions: 6 × 10, 5 × 12, 4 × 15, and 3 × 20. (A 2 × 30 rectangle would be too narrow to accommodate all the shapes.)

Clarke, who rarely played intellectual games, found that this challenge “can rather rapidly escalate — if you have that sort of mind — into a way of life.” He stole a set of tiles from his niece, spent hundreds of hours playing with it, and even worked the shapes into the design of a rug for his office. “That a jigsaw puzzle consisting of only 12 pieces cannot be quickly solved seems incredible, and no one will believe it until he has tried,” he wrote in the Sunday Telegraph Magazine. It took him a full month to arrange the 12 shapes into a 6 × 10 rectangle — a task that he was later abashed to learn can be done in 2339 different ways. There are 1010 solutions to the 5 × 12 rectangle and 368 solutions to the 4 × 15.
Image: Wikimedia Commons

But “The most interesting case, however, is that of the long, thin rectangle only 3 units wide and 20 long.” Clarke became fascinated with this challenge when Martin Gardner revealed that only two solutions exist. He offered 10 rupees to anyone who could find the solutions, and was delighted when a friend produced them, as he’d calculated that solving the problem by blind permutation would take more than 20 billion years.

Clarke even worked the 3 × 20 problem into his 1975 novel Imperial Earth. Challenged by his grandmother, the character Duncan struggles with the task and declares it impossible. “I’m glad you made the effort,” she says. “Generalizing — exploring every possibility — is what mathematics is all about. But you’re wrong. It can be done. There are just two solutions; and if you find one, you’ll also have the other.”

Can you find them?

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