“Another Paradox”

http://books.google.com/books?id=ehgDAAAAYAAJ&printsec=frontcover&source=gbs_slider_thumb#v=onepage&q&f=false

If a cork ball about an inch in diameter be tied at the end of a thread about a foot in length, and then swung so that it enters a smooth stream of water flowing from a tap at about three inches from the mouth of the latter, it will be found that the ball will remain in the water, and that the thread will make an angle of about thirty degrees with a vertical line passing through the ball. The latter, it should be added, must be thoroughly wetted before this result is produced.

Strand, September 1908

Math Notes

20864448472975628947226005981267194447042584001 = (2 + 0 + 8 + 6 + 4 + 4 + 4 + 8 + 4 + 7 + 2 + 9 + 7 + 5 + 6 + 2 + 8 + 9 + 4 + 7 + 2 + 2 + 6 + 0 + 0 + 5 + 9 + 8 + 1 + 2 + 6 + 7 + 1 + 9 + 4 + 4 + 4 + 7 + 0 + 4 + 2 + 5 + 8 + 4 + 0 + 0 + 1)20

Moving Without Motion?

Achilles-weed is prostrate and grows along the ground at the amazing rate of 10 cm per hour. An exceeding slow tortoise munches one end of the Achilles-weed at the same rate as it grows at the other end. So the tortoise appears to chase the Achilles-weed round the garden. But, strictly speaking, the Achilles-weed does not move at all, it grows and is eaten. Yet its location changes, and it is made up of parts whose location changes (the left and right-hand halves of the Achilles-weed). Hence being made up of parts whose location changes is not sufficient for motion.

— Peter Forrest, “Is Motion Change of Location?”, Analysis, 1984

A Pretty Problem

In Longfellow’s novel Kavanagh, Mr. Churchill reads a word problem to his wife:

“In a lake the bud of a water-lily was observed, one span above the water, and when moved by the gentle breeze, it sunk in the water at two cubits’ distance. Required the depth of the water.”

“That is charming, but must be very difficult,” she says. “I could not answer it.”

Is it? If a span is 9 inches and a cubit is 18 inches, how deep is the water?

Click for Answer

Misc

  • SWEET-TOOTHED has three consecutive pairs of letters. SUBBOOKKEEPER has four.
  • Will you answer this question negatively?
  • 4624 = 44 + 46 + 42 + 44
  • The telephone number 278-7433 spells both ASTRIDE and CRUSHED.
  • “Nothing great was ever achieved without enthusiasm.” — Emerson

Beach Reading

Amazon reviews of A Million Random Digits with 100,000 Normal Deviates (1955), by the RAND Corporation:

  • “I had a hard time getting into this book. The profanity was jarring and stilted, not at all how people really talk.”
  • “Once you get about halfway in, the rest of the story is pretty predictable.”
  • “If you like this book, I highly recommend that you read it in the original binary.”
  • “I would have given it five stars, but sadly there were too many distracting typos. For example: 46453 13987.”
  • “I really liked the ‘10034 56429 234088’ part.”
  • “Frankly the sex scenes were awkward and clumsily written, adding very little of value to the plot.”
  • “For a supposedly serious reference work the omission of an index is a major impediment. I hope this will be corrected in the next edition.”

The average customer gives it four stars.

Misc

  • EVIAN, SEIKO, and STROH’S are all English words spelled backward.
  • Can “I apologize” be false?
  • 165033 = 163 + 503 + 333
  • Little Wymondley, in Hertfordshire, is bigger than Great Wymondley.
  • “How old would you be if you didn’t know how old you was?” — Satchel Paige

A Surprise Visitor

Create two columns, one starting with the numbers 12 and 18 and the other with 5 and 5. Continue each column, deriving each new number by adding the two that precede it:

JRM pi ratios

In the Journal of Recreational Mathematics, James Davis writes, “Forming successive pairs with adjoining numbers from each column one finds the ratio of the two numbers in each pair converges to π!” How can this be?

“The alert reader will suspect there is a trick in this method, as I did when π first presented it to me. The labor of several hours of computation coupled with trial and error produced half of the secret of the method. It is obviously based somehow on the fact that φ (the golden mean, which equals \frac{1+\sqrt{5}}{2}, can be closely approximated by the nifty pseudo equation below:”

1.2 \times \varphi ^{2}=\pi

“Can the reader decipher π’s technique for making herself with φ?”