The Value of Disagreement

In 1907, Francis Galton famously found that when a crowd were asked to guess the weight of an ox, the average value of their responses was surprisingly accurate — in Galton’s experiment, it fell within 1 percent of the ox’s true weight. This is “the wisdom of crowds”: By canceling errors across individuals, the mean response often proves more accurate than individual estimates.

Interestingly, the same phenomenon can arise when we aggregate multiple estimates made by a single person (the “wisdom of the inner crowd”). And organizational behavior researchers Philippe van de Calseyde and Emir Efendić now find that the accuracy can be refined still further when people are asked to consider a question from the perspective of someone they often disagree with.

“In explaining its accuracy, we find that taking a disagreeing perspective prompts people to consider and adopt second estimates they normally would not consider as viable option, resulting in first and second estimates that are highly diverse (and by extension more accurate when aggregated),” the researchers write. “Our results suggest that disagreement, often highlighted for its negative impact, can be a powerful tool in producing accurate judgments.”

(Philippe van de Calseyde and Emir Efendić, “Taking a Disagreeing Perspective Improves the Accuracy of People’s Quantitative Estimates,” PsyArXiv, Nov. 15, 2019.)

December 8, 2020December 7, 2020 | Science & Math · Society

Unquote

“It tires me to talk to rich men. You expect a man of millions, the head of a great industry, to be a man worth hearing; but as a rule they don’t know anything outside their own businesses.” — Theodore Roosevelt

December 7, 2020December 5, 2020 | Quotations · Society

Podcast Episode 322: Joseph Medicine Crow

Joseph Medicine Crow was raised on a Montana reservation in the warrior tradition of his Crow forefathers. But during World War II he found himself applying those lessons in very different circumstances. In this week’s episode of the Futility Closet podcast, we’ll describe Joseph’s exploits in the war and how they helped to shape his future.

We’ll also consider how to distinguish identical twins and puzzle over a physicist’s beer.

See full show notes …

December 7, 2020December 14, 2020 | History · Podcast

The Substitute

Between July 2015 and October 2018, the National Security Agency offered a monthly puzzle written by a staff member. Here’s the puzzle for October 2015, created by applied research mathematician Ben E.:

Kurt, a math professor, has to leave for a conference. At the airport, he realizes he forgot to find a substitute for the class he was teaching today! Before shutting his computer off for the flight, he sends an email: “Can one of you cover my class today? I’ll bake a pie for whomever can do it.” He sends the email to Julia, Michael, and Mary Ellen, his three closest friends in the math department, and boards the plane.

As Kurt is well-known for his delicious pies, Julia, Michael, and Mary Ellen are each eager to substitute for him. Julia, as department chair, knows which class Kurt had to teach, but she doesn’t know the time or building. Michael plays racquetball with Kurt so he knows what time Kurt teaches, but not the class or building. Mary Ellen helped Kurt secure a special projector for his class, so she knows what building Kurt’s class is in, but not the actual class or the time.

Julia, Michael, and Mary Ellen get together to figure out which class it is, and they all agree that the first person to figure out which class it is gets to teach it (and get Kurt’s pie). Unfortunately the college’s servers are down, so Julia brings a master list of all math classes taught that day. After crossing off each of their own classes, they are left with the following possibilities:

Calc 1 at 9 in North Hall
Calc 2 at noon in West Hall
Calc 1 at 3 in West Hall
Calc 1 at 10 in East Hall
Calc 2 at 10 in North Hall
Calc 1 at 10 in South Hall
Calc 1 at 10 in North Hall
Calc 2 at 11 in East Hall
Calc 3 at noon in West Hall
Calc 2 at noon in South Hall

After looking the list over, Julia says, “Does anyone know which class it is?” Michael and Mary Ellen immediately respond, “Well, you don’t.” Julia asks, “Do you?” Michael and Mary Ellen both shake their heads. Julia then smiles and says, “I do now. I hope he bakes me a chocolate peanut butter pie.”

Which class does Kurt need a substitute for?

	Select Click for Answer
1) Since Julia only knows the class name, the only way she could immediately know is if it was Calc 3. Since Michael and Mary Ellen both know that Julia doesn’t know, that means they know the class isn’t Calc 3. Since Michael only knows the time, that means the class can’t be at noon. Because Mary Ellen only knows the building, that means the building can’t be West Hall. That leaves only the following possibilities: Calc 1 at 9 in North Hall Calc 1 at 10 in East Hall Calc 2 at 10 in North Hall Calc 1 at 10 in South Hall Calc 1 at 10 in North Hall Calc 2 at 11 in East Hall 2) Since Michael doesn’t know which class it is, that means the time can’t be 9 or 11. Since Mary Ellen doesn’t know either, the class can’t be in South Hall. That leaves only three possibilities: Calc 1 at 10 in East Hall Calc 2 at 10 in North Hall Calc 1 at 10 in North Hall 3) At this point, Julia now says she knows the answer. Since there are two Calc 1 classes, it must be that Julia knows the class is Calc 2. Thus, the class is Calc 2 at 10 in North Hall.

Click for Answer

December 6, 2020 | Puzzles

Quarters

USC mathematician Solomon Golomb offered this puzzle in his column, “Golomb’s Gambits,” in Johns Hopkins Magazine. How can you dissect this figure into four congruent pieces?

	Select Click for Answer
Donald Savage found this solution in a computer search described in the July-August 1999 issue of MIT Technology Review. It appears to be unique. 12/08/2020 UPDATE Not unique! Reader Irit Rubin came up with this alternate solution: (Thanks, Irit.)

December 5, 2020December 8, 2020 | Puzzles

Old Booty’s Ghost

A striking tale from the 18th century: It’s said that around 1687 a group of English mariners on the Italian coast were surprised to see “two men run by us with amazing swiftness”:

Captain Barnaby says, ‘Lord bless me, the foremost man looks like next door neighbour, old Booty;’ but said he did not know the other behind. Booty was dressed in grey clothes, and the one behind him in black; we saw them run into the burning mountain in the midst of the flames! on which we heard a terrible noise, too horrible to be described.

When they returned to Gravesend, Captain Barnaby’s wife said, “My dear, I have got some news to tell you; old Booty is dead.” Barnaby swore an oath and said, “We all saw him run into Hell!”

As the story goes, when word of this allegation reached Booty’s widow, she sued Barnaby for a thousand pounds. The punchline is that Booty’s appearance on the volcano was shown to have occurred within two minutes of his death, and when his coat was exhibited in the courtroom, 12 sailors swore that its buttons matched those of the fleeing man.

The Judge then said, ‘Lord grant I may never see the sight that you have seen; one, two, or three may be mistaken, but twenty or thirty cannot.’ So the widow lost her cause.

According to folklorist Jeremy Harte, this story appeared in print at least 19 times between the 1770s and the 1830s. It seems to have started among the dockyards of the lower Thames, where in one early version Booty was an unscrupulous contractor who had supplied the navy with adulterated beer — and his damnation was “a matter of just retribution for the sin he had committed.”

(Jeremy Harte, “Into the Burning Mountain: Legend, Literature, and Law in Booty v. Barnaby,” Folklore 125:3 [December 2014], 322-338.)

December 4, 2020December 4, 2020 | Oddities

After You

A problem from Canada’s 2003 Hypatia contest:

Xavier and Yolanda are playing a game. They begin with two piles of three coins each and take turns; on each turn a player removes one or more coins from any one pile. The winner is the player who takes the very last coin. Xavier always goes first, but Yolanda has a strategy that ensures that she will always win. What is it?

	Select Click for Answer
She emulates each of Xavier’s moves in the opposite pile. If he takes one coin from Pile 1 then she takes one coin from Pile 2, and so on. This forces him to empty one pile first, whereupon she can empty the other. (Via Robert Bilinski, “Skoliad,” Crux Mathematicorum 31:1 [January 2005], 1-12.)

December 4, 2020 | Puzzles

“Half-Hanged Smith”

John Smith escaped execution three times. Convicted of housebreaking in 1705, he was hanged at the Tyburn gallows for a quarter of an hour before the people called for a reprieve and he was cut down.

When he had perfectly recovered his senses he was asked what were his feelings at the time of execution; to which he repeatedly replied, in substance, as follows. When he was turned off, he for some time was sensible of very great pain, occasioned by the weight of his body, and felt his spirits in a strange commotion, violently pressing upwards. That having forced their way to his head, he as it were saw a great blaze, or glaring light, which seemed to go out at his eyes with a flash, and then he lost all sense of pain. That after he was cut down, and began to come to himself, the blood and spirits, forcing themselves into their former channels, put him, by a sort of pricking or shooting, to such intolerable pain that he could have wished those hanged who had cut him down.

He returned to housebreaking on his release in 1706, but his strange luck continued: On his second indictment some difficulties in the case induced a panel of judges to set him free, and on his third the prosecutor died before the day of the trial.

The streak ended in 1727, when he was convicted of stealing a padlock and sentenced to transportation. He pleaded for corporal punishment instead but was sent to Virginia that July.

December 3, 2020December 2, 2020 | Crime

Constrained Travel

In 1853, a lady at a social dinner challenged explorer Jacques Arago to compose an account of his circumnavigation without using the letter A.

He did.

Allegedly she thanked him without using the letter C.

December 3, 2020December 2, 2020 | Language

Misc

Peter Davison, who played the fifth Doctor in Doctor Who, is the father-in-law of David Tennant, who played the 10th.
Sharks are older than trees.
ABHORS, ALMOST, BEGINS, BIOPSY, and CHINTZ are alphabetical.
$\displaystyle \sqrt{7! + 1} = 71$
“The punishment can be remitted; the crime is everlasting.” — Ovid

“Whoopee! Man, that may have been a small one for Neil, but that’s a long one for me!” — Pete Conrad, after becoming the third human to set foot on the moon

December 2, 2020December 2, 2020 | Entertainment · History · Language · Quotations · Science & Math · Trivia