Fruitful Dreams

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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.”

| Science & Math · Technology

Non/Fiction

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?

| Science & Math

Straight and Narrow

http://commons.wikimedia.org/wiki/File:Monge_theorem.svg

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
| Puzzles · Science & Math

Piecemeal

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

| Science & Math

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
| Puzzles · Science & Math

Quite Contrary

A contronym is a word with two contrary meanings, such as cleave or sanction (more here).

The word contronym itself has no double meaning. Is it a contronym?

“Not until I came along!” writes Charles Melton in Word Ways. “I declare that it is a contronym for the simple reason that it isn’t! It is both a self-opposite and not a self-opposite. QED.”

| Language · Science & Math

A Violent Argument

A logic exercise by Lewis Carroll. What conclusion is implied by these premises?

  1. Animals that do not kick are always unexcitable.
  2. Donkeys have no horns.
  3. A buffalo can always toss one over a gate.
  4. No animals that kick are easy to swallow.
  5. No hornless animal can toss one over a gate.
  6. All animals are excitable except buffaloes.
| Science & Math

Bang!

If Brown hopes to throw a six in a game of dice and succeeds, we wouldn’t say he threw the six intentionally. If Brown puts his last cartridge into a six-chambered revolver, spins the chamber as he aims it at Smith, his archenemy, pulls the trigger, and kills Smith, we’d say he killed him intentionally. Does that make sense? In both cases Brown hoped for a certain result, in both cases the probability of that result was the same. If Brown didn’t intentionally throw a six, why did he intentionally shoot Smith?

— Leo Katz, Bad Acts and Guilty Minds, 1987

| Science & Math · Society

Misc

  • What time is it on the sun?
  • PATERNAL, PARENTAL, and PRENATAL are anagrams.
  • If forecastle is pronounced “fo’c’sle,” should forecast be pronounced “folks”?
  • A clock’s second hand is its third hand.
  • “The religion of one seems madness unto another.” — Thomas Browne

Bonus poser: In what sport does only the winning team travel backward?

| Language · Quotations · Science & Math