“A Geometrical Paradox”

A stick is broken at random into 3 pieces. It is possible to put them together into the shape of a triangle provided the length of the longest piece is less than the sum of the other 2 pieces; that is, provided the length of the longest piece is less than half the length of the stick. But the probability that a fragment of a stick shall be half the original length of the stick is 1/2. Hence the probability that a triangle can be constructed out of the 3 pieces into which the stick is broken is 1/2.

— Samuel Isaac Jones, Mathematical Wrinkles, 1912

(The actual probability is 1/4.)

Aliquot Sequences

Pick a positive integer, list the positive integers that will divide it evenly, add these up, and subtract the number itself:

  • 10 is evenly divisible by 10, 5, 2, and 1. (10 + 5 + 2 + 1) – 10 = 8.

Now do the same with that number, and continue:

  • 8 is evenly divisible by 8, 4, 2, and 1. (8 + 4 + 2 + 1) – 8 = 7.
  • 7 is evenly divisible by 7 and 1. (7 + 1) – 7 = 1.
  • 1 is evenly divisible only by 1. (1) – 1 = 0.

Many of these sequences arrive at some resolution — they terminate in a constant, or an alternating pair, or some regular cycle. But it’s an open question whether all of them do this. The fate of the aliquot sequence of 276 is not known; by step 469 it’s reached 149384846598254844243905695992651412919855640, but possibly it reaches some apex and then descends again and finds some conclusion (the sequence for the number 138 reaches a peak of 179931895322 but eventually returns to 1). Do all numbers eventually reach a resolution? For now, no one knows.

The Egyptian Lo Shu

https://commons.wikimedia.org/wiki/File:Magic_Square_Lo_Shu.svg

Another contribution from Lee Sallows:

“The smallest, oldest and most famous magic square of all is the specimen of Chinese origin known as the Lo shu. In this, the numbers from 1 to 9 are so placed that their sum taken in any row, column or diagonal is 15. This is another way of saying that the sum of any three of them lying in a straight line is 15. Less well known is the ‘Egyptian’ Lo shu (seen below) in which the same numbers are rearranged in a triangular formation that exhibits the same property.”

(From his book Geometric Magic Squares, 2013.) (Thanks, Lee.)

sallows egyptian lo shu

Podcast Episode 252: The Wild Boy of Aveyron

https://commons.wikimedia.org/wiki/File:Jean_marc_gaspard_itard_1775_hi.jpg

In 1800 a 12-year-old boy emerged from a forest in southern France, where he had apparently lived alone for seven years. His case was taken up by a young Paris doctor who set out to see if the boy could be civilized. In this week’s episode of the Futility Closet podcast we’ll explore the strange, sad story of Victor of Aveyron and the mysteries of child development.

We’ll also consider the nature of art and puzzle over the relationship between salmon and trees.

See full show notes …

The Taxicab Problem

A cab was involved in a hit and run accident at night. Two cab companies, the Green and the Blue, operate in the city. 85% of the cabs in the city are Green and 15% are Blue.

A witness identified the cab as Blue. The court tested the reliability of the witness under the same circumstances that existed on the night of the accident and concluded that the witness correctly identified each one of the two colors 80% of the time and failed 20% of the time.

What is the probability that the cab involved in the accident was Blue rather than Green knowing that this witness identified it as Blue?

Psychologists Amos Tversky and Daniel Kahneman offered this problem to study subjects in 1972. The right answer is about 41 percent:

  • There’s a 12% chance (15% times 80%) of the witness correctly identifying a blue cab.
  • There’s a 17% chance (85% times 20%) of the witness incorrectly identifying a green cab as blue.
  • Thus there’s a 29% chance (12% plus 17%) that the witness will identify the cab as blue.
  • And that means there’s approximately a 41% chance (12% divided by 29%) that the cab identified as blue is really blue:

https://en.wikipedia.org/wiki/File:TaxicabProblem.png

Most subjects estimated the probability at more than 50 percent, some more than 80 percent.

Tversky and Kahneman call this the representativeness heuristic: When we rely on representativeness to make a judgment, we tend to judge wrongly because the fact that a thing is more representative doesn’t make it more likely.

(Amos Tversky and Daniel Kahneman, “Evidential Impact of Base Rates,” No. TR-4, Stanford University Department of Psychology, 1981.)

The Perpetual Diamond

This is bewildering: This diamond isn’t moving, and its luminance and texture are unchanging. Yet when it’s surrounded with very thin edge strips whose luminance changes with respect to the background, the whole diamond seems to move. Using the controls at the bottom, you can even direct the illusion to send the diamond drifting “up,” “down,” “left,” or “right.” But it ain’t moving.

See the paper below for details.

(Oliver J. Flynn and Arthur G. Shapiro, “The Perpetual Diamond: Contrast Reversals Along Thin Edges Create the Appearance of Motion in Objects,” i-Perception 9:6 [2018], 2041669518815708.)

Misc

https://www.flickr.com/photos/pondapple/16890700391
Image: Flickr
  • At the equinox, the sun rises due east at every latitude.
  • UPPER TYPEWRITER ROW is typed on the upper row of a typewriter.
  • 32785 = 3 + 2 × 7 + 85
  • In the Mbabaram Aboriginal language of north Queensland, dog means dog.
  • The London Times has published no obituary for Sherlock Holmes. Therefore he exists.

(Thanks, Sanford.)

The McGurk Effect

In 1976 psychologist Harry McGurk discovered that seeing a person speak affects our impression of the sound we hear. Faced with conflicting information, the brain seems to make its “best guess” as to what it’s perceiving. In some cases a third sound is produced: When the syllables /ba-ba/ are spoken over the lip movements /ga-ga/, the perception is /da-da/.

This casts doubt on the assumption that the senses operate separately and can be studied in isolation. Psychologists and philosophers are still considering the implications.

(Harry McGurk and John MacDonald, “Hearing Lips and Seeing Voices,” Nature 264:5588 [1976], 746.)

The False Position Method

In David Hayes and Tatiana Shubin’s Mathematical Adventures (2004), University of California-Davis mathematician Don Chakerian describes a method used in antiquity for solving an equation in one unknown. He illustrates it with a problem from Daboll’s Schoolmaster’s Assistant (1800):

A, B, and C built a house which cost $500, of which A paid a certain sum, B paid 10 dollars more than A, and C paid as much as A and B both; how much did each man pay?

We’ll make two guesses as to how much A paid, check them, and plug the “errors” into a formula to get the right answer. First, suppose A pays $80. That means that B pays $90 and C pays $170, giving a total of $340. That’s 500 – 340 = $160 short of the goal, so our guess of $80 yields an “error” of $160. As a second guess, suppose that A pays $150. In that case B pays $160, C pays $310, and the total is now $620. This time the “error” is 500 – 620 = -$120. The false position method (technically here the double false position method) offers this formula for finding the right answer:

\displaystyle \frac{\left ( \textup{first guess} \right ) \left ( \textup{second error} \right ) - \left ( \textup{second guess} \right ) \left ( \textup{first error} \right )}{\left ( \textup{second error} \right ) - \left ( \textup{first error} \right )}

In this case it gives

\displaystyle \frac{\left ( 80 \right ) \left ( -120 \right ) - \left ( 150 \right ) \left ( 160 \right )}{ -120   -160 } = \frac{-9600 - 24000}{ -280 } = 120.

When A pays $120 then B pays $130, C pays 250, and together they pay $500, so this solution works.

This is hardly the most efficient way to solve a simple linear equation given the tools we have today, but it served for centuries. In his Ground of Artes of 1542, Robert Recorde offered a rule:

Gesse at this woorke as happe doth leade.
By chaunce to truthe you may procede.
And firste woorke by the question,
Although no truthe therein be don.
Suche falsehode is so good a grounde,
That truth by it will soone be founde.
From many bate to many mo,
From to fewe take to fewe also.
With to much ioyne to fewe againe,
To to fewe adde to manye plaine.
In crossewaies multiplye contrary kinde,
All truthe by falsehode for to fynde.

Many-Sided Story

From Ed Southall’s Twitter feed, a polygon name builder:

https://twitter.com/solvemymaths/status/1118049237273649152

A 55-sided figure is a pentacontapentagon; one with 79 sides is a heptacontaenneagon. A system exists to go even higher: A figure with 9,000 sides is an enakischiliagon, and one with a million is a megagon.

René Descartes suggested the 1,000-sided chiliagon as an example of a thing that can be considered without being explicitly imagined; one “does not imagine the thousand sides or see them as if they were present.” So the intellect is not dependent on imagination.