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.
Killed by an omnibus — why not?
So quick a death a boon is.
Let not his friends lament his lot —
Mors omnibus communis.
— Henry Luttrell
(Mors omnibus communis means “Death is common to all men.”)
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.)
n. the feeding or fattening of pigs
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.
This Victorian artificial arm and hand is in the collection of the London Science Museum. “The arm is amazingly versatile,” writes Ben Russell in Robots (2017). “The elbow can be locked in several positions, and the fingers articulated using a brass button in the wrist. It is also heavily decorated in the neo-Gothic style. Rather than being covered up, this arm would be out on view, making its wearer a true man-machine.”
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.)
Did Ophelia ask Hamlet to bed?
Was Gertrude incestuously wed?
Is there anything certain?
By the fall of the curtain
Almost everyone’s certainly dead.
— A. Cinna
Once a raven on Pluto’s dark shore
Brought the singular news: “Nevermore.”
‘Twas of useless avail
To ask further detail,
His reply was the same as before.
— Anthony Euwer
There once was a fellow called Hyde,
Whose twin self he couldn’t abide;
But Jekyll, the Devil,
Dragged Hyde to his level,
“Inside job,” cried Hyde, as he died.
— E.J. Jackson
When Ireland was bloody and leaderless,
The tedious, garrulous Daedalus —
Having failed both as priest
And as Glorious Beast —
Sailed away to write books that were readerless.
— Gina Berkeley
Is it possible to divide a circle into n parts of equal area using only a straightedge and compasses?