A Hat Puzzle

Each of three people is wearing either a red hat or a blue hat. Each can see the color of the others’ hats but not her own. Each is told to raise her hand if she sees a red hat on another player. The first to guess the color of her own hat correctly wins.

All three raise their hands. A few minutes pass in which no guesses are made, and then one player says “Red” and wins. How did she know the color of her hat?

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Mercyhurst College mathematician Charles Redmond used to get frantic phone calls at the end of each term from colleagues in other departments who’d written syllabi saying “Quiz 1 is 10% of your final grade” and now couldn’t figure out how to do the necessary calculation.

“There’s a good reason,” Redmond wrote. “They said something they didn’t mean to say, and they had only a vague notion of what they meant to say in the first place.”

He suggested that students might turn this confusion to their advantage. Suppose Quiz 1 is worth 10 or more points and your score is 9. If x is your final grade and “Quiz 1 is 10% of your final grade,” then 0.10x = 9 and suddenly your final grade is 90, an A.

“Not bad for about a week’s worth of work. Take the rest of the semester off.”

(Ed Barbeau, “Fallacies, Flaws, and Flimflam,” College Mathematics Journal 33:2 [March 2002], 137-139.)

Another Way

Carnegie-Mellon mathematician Po-Shen Loh has offered a new, simple proof of the quadratic formula that provides a natural, intuitive algorithm for solving general quadratic equations.

More here. “May this story encourage the reader to think afresh about old things; seeing as how new progress was made on this 4,000 year old topic, more surprises certainly await the light of discovery.”

(Thanks, Jason.)

Podcast Episode 276: An Unlikely Confederate Spy


As the Civil War fractured Washington D.C., socialite Rose O’Neal Greenhow coordinated a vital spy ring to funnel information to the Confederates. In this week’s episode of the Futility Closet podcast we’ll describe one of the war’s most unlikely spies, and her determination to aid the South.

We’ll also fragment the queen’s birthday and puzzle over a paid game of pinball.

See full show notes …

First Friends

In 1978, archaeologists excavating a late Paleolithic tomb in northern Israel uncovered the skeletons of an elderly human and a 5-month-old puppy. They had lain there together for 12,000 years.

“The most striking thing about these remains was the fact that whoever presided over the original burial had carefully arranged the dead person’s left hand so that it rested, in a timeless and eloquent gesture of attachment, on the puppy’s shoulder,” writes James Serpell in In the Company of Animals (1996).

“The contents of this tomb not only provide us with some of the earliest solid evidence of animal domestication, they also strongly imply that man’s primordial relationship with this particular species was an affectionate one. In other words, prehistoric man may have loved his dogs and his other domestic animals as pets long before he made use of them for any other purpose.”

(Simon J.M. Davis and François R. Valla, “Evidence for Domestication of the Dog 12,000 Years Ago in the Natufian of Israel,” Nature 276:5688 [1978], 608.)

The Friendly Floatees

Image: Wikimedia Commons

During a storm in January 1992, a container was swept overboard from a ship in the North Pacific. As it happened, it contained 28,800 children’s bath toys, and oceanographer Curtis Ebbesmeyer realized they offered the basis for a serendipitous study of surface currents. Working with his colleague James Ingraham, Ebbesmeyer began to track the toys as they drifted around the globe, accumulating reports from beachcombers, coastal workers, and local residents as they began to wash up on beaches. Using computer models, they were able to predict correctly that toys would make landfall in Washington state, Japan, and Alaska, and even become trapped in pack ice and spend years creeping across the top of the world before making an eventual reappearance in the North Atlantic. “Ultimately,” Ebbesmeyer wrote, “the toys will turn to dust, joining the scum of plastic powder which rides the global ocean.”

For some reason, media accounts of the story always carried the image of a solitary rubber duck, though the toys had also included beavers, turtles, and frogs. “Maybe it’s a kind of racism,” Ebbesmeyer speculated to journalist Donovan Hohn in 2007. “Speciesism.”