Asking Directions

If we take a cube and label one side top, another bottom, a third front, and a fourth back, there remains no form of words by which we can describe to another person which of the remaining sides is right and which left. We can only point and say here is right and there is left, just as we should say this is red and that blue.

— William James, The Principles of Psychology, 1890

Hose and Cons

Sir John Cutler had pair of silk stockings, which his housekeeper, Dolly, darned for a long term of years with worsted; at the end of which time, the last gleam of silk had vanished, and Sir John’s silk stockings were found to have degenerated into worsted. Now, upon this, a question arose amongst the metaphysicians, whether Sir John’s stockings retained (or, if not, at what precise period they lost) their personal identity. The moralists again were anxious to know, whether Sir John’s stockings could be considered the same ‘accountable’ stockings from first to last. The lawyers put the same question in another shape, by demanding whether any felony which Sir John’s stockings could be supposed to have committed in youth, might legally be the subject of indictment against the same stockings when superannuated; whether a legacy left to the stockings in their first year, could be claimed by them in their last; and whether the worsted stockings could be sued for the debts of the silk stockings.

— Thomas de Quincey, “Autobiography of an English Opium-Eater,” from Tait’s Edinburgh Magazine, September 1838


J.J. Sylvester was a brilliant mathematician but, by all accounts, a lousy poet. The Dictionary of American Biography opines delicately that “Most of Sylvester’s original verse showed more ingenuity than poetic feeling.”

What it lacked, really, was variety. His privately printed book Spring’s Debut: A Town Idyll contains 113 lines, every one of which rhymes with in.

Even worse is “Rosalind,” a poem of 400 lines all of which rhyme with the title character’s name. In his History of Mathematics, Florian Cajori reports that Sylvester once recited “Rosalind” at Baltimore’s Peabody Institute. He began by reading all the explanatory footnotes, so as not to interrupt the poem, and realized too late that this had taken an hour and a half.

“Then he read the poem itself to the remnant of his audience.”

See Poetry in Motion.

C Sickness

“Light crosses space with the prodigious velocity of 6,000 leagues per second.”

La Science Populaire, April 28, 1881

“A typographical error slipped into our last issue that it is important to correct: the speed of light is 76,000 leagues per hour — and not 6,000.”

La Science Populaire, May 19, 1881

“A note correcting a first error appeared in our issue number 68, indicating that the speed of light is 76,000 leagues per hour. Our readers have corrected this new error. The speed of light is approximately 76,000 leagues per second.”

La Science Populaire, June 16, 1881

Fruitful Dreams

In 1862, August Kekulé dreamed of a snake seizing its own tail; the vision inspired him to propose the structure of the benzene molecule.

Louis Agassiz had been struggling for two weeks to decipher the impression of a fossil fish in a stone slab when he dreamed on three successive nights of its proper character. When he chiseled away the stone he found that the hidden portions of the fish matched his nocturnal drawing.

William Watts had been forming lead shot mechanically when he dreamed he was caught in a cloudburst of molten metal. The image inspired him to develop the shot tower.

The best such story, alas, is false. It’s said that Elias Howe, frustrated in devising a sewing machine, dreamed he had been captured by an African tribe. He noticed that the menacing warriors’ spear-tips bore holes, and this inspired him to move the hole in his machine’s needle from the dull end (as in a hand needle) to the sharp one.

“This is not true,” writes Alonzo Bemis. “Mr. Howe was too much of a Yankee to place any dependence in dreams, and the needle idea was worked out by careful thought and countless experiments.”


Alfred Tarski imagines a 100-page book in which page 1 reads, “The statement on page 2 of this book is true.” Page 2 reads, “The statement on page 3 of this book is true.” This continues until page 100, which reads, “The statement on page 1 of this book is false.” Is the statement on page 67 true or false?

In writing the preface for a new book, an author commonly thanks those who helped him and concludes, “I am responsible for the inevitable errors that remain.” David Makinson notes that the author now seems to believe, simultaneously and rationally, that each given statement in the book is accurate and that at least one of them isn’t.

(William Poundstone notes that the author might try to escape this problem by writing instead, “At least one of the statements in this book is false.” Now if the text itself is clean, the disclaimer cancels itself … or does it?)

Kurt Vonnegut’s 1963 novel Cat’s Cradle is prefaced with the statement “Nothing in this book is true.” Is this statement true?

Straight and Narrow

Draw three nonintersecting circles of different sizes, and bracket each pair of them with tangents. Each pair of tangents will intersect in a point, and these three points will always lie along a line.

On being shown this theorem, Cornell engineering professor John Edson Sweet paused and said, “Yes, that is perfectly self-evident.” What intuitive proof had he seen?

Click for Answer


The following successful though illegal operation is reported by Professor A. Gloden in Sphinx, Volume VI, Number 7. In multiplying 6 2/3 by 4 4/5 a student first found the product of the integers, 6 × 4 = 24. He then reduced the fractions to the common denominator … and divided the product of the numerators 10 × 12 by the common denominator 15. The result 24 + 8 = 32 is correct.

In the same way he obtained the correct value of the product 9 3/5 × 2 2/4.

— “Curiosa,” Scripta Mathematica, October 1936

Checkered Doughnuts

toroidal magic square - from mathematical circles

Roll this magic square into a tube by joining the upper and lower edges, then join the ends of the tube. Every row, column, and diagonal on the resulting torus will add to 34.

toroidal chess problem, from petrovic, mathematics and chess

Bend this chessboard similarly into a torus, then mate in 4.

Hint: The solution comprises only two lines.

Click for Answer