The Chrysler Norseman

For Chrysler’s 1957 auto show, designer Virgil Exner prepared a one-of-a-kind prototype: the Norseman, a sleek four-seat fastback coupe with a sloping hood, cantilevered roof, and aerodynamic underbody.

After 15 months’ work, the fully drivable $150,000 concept car missed its shipment date and was put aboard the next available transport.

That was the SS Andrea Doria. The unique prototype was lost in the sinking, and the car was never produced.

Grammatical Illusions

More people have been to Russia than I have.

Most listeners find this sentence acceptable when they first hear it, but it’s meaningless: The phrase “more people” seems to set up a comparison between two sets of individuals, but there’s no second set.

“In light of the fact that the sentence lacks this basic property, it is remarkable that speakers so commonly fail to notice the error,” write linguists Colin Phillips, Matthew W. Wagers, and Ellen F. Lau.

No head injury is too trivial to be ignored.

At first this seems to mean “No head injury should be ignored — even if it’s trivial,” but reflection shows that it really means “All head injuries should be ignored — even trivial ones.”

“This difficulty has certain interesting properties,” write psychologists Peter Wason and Shuli Reich. “When the correct interpretation was explained it was often adamantly rejected in our informal studies, as if the informants literally could not see an alternative view.”

(Colin Phillips, M. Wagers, and E. Lau, “Grammatical Illusions and Selective Fallibility in Real-Time Language Comprehension,” in Jeffrey T. Runner, ed., Syntax and Semantics 37: Experiments at the Interfaces, 2011; Peter C. Wason and Shuli S. Reich, “A Verbal Illusion,” Quarterly Journal of Experimental Psychology 31:4 [1979], 591-597.)

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.)


“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

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 …

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?

Click for Answer


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?

Click for Answer

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.)

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?

Click for Answer