# Thought

“It is interesting that most of the human race has a reserve of the enzyme necessary to render alcohol harmless to the body — as if nature meant us to drink alcohol, unlike animals to which alcohol is a poison.”

BUPA News, 1982, quoted in Richard Gordon, Great Medical Mysteries, 2014

# Van der Waerden’s Theorem

Number eight cells:

Now suppose we want to color each cell red or blue such that no three cells are in arithmetic progression — for example, we don’t want cells 1, 2, and 3 to be the same color, or 4, 6, and 8. With eight cells it’s possible to accomplish this:

But if we want to add a ninth cell we can’t avoid an arithmetic progression: If the ninth cell is blue then cells 1, 5, and 9 are evenly spaced, and if it’s red then cells 3, 6, and 9 are. Dutch mathematician B.L. van der Waerden found that there’s always such a limit: For any given positive integers r and k, there’s some number N such that if the integers {1, 2, …, N} are colored, each with one of r different colors, then there will be at least k integers in arithmetic progression whose elements are of the same color. Determining what this limit is (in this example it’s 9) is an open problem.

(Bonus: Alexej Kanel-Belov found this pretty theorem concerning divisibility of integer sums within an infinite grid — Martin J. Erickson, in Beautiful Mathematics, calls it a two-dimensional version of van der Waerden’s theorem.)

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

# 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

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.)

# Podcast Episode 252: The Wild Boy of Aveyron

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:

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

• 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.)