Math Notes

The American Mathematical Monthly of January 1959 notes an “interesting Pythagorean triangle” discovered by Victor Thébault: If the two perpendicular sides of a right triangle measure 88209 and 90288, then the hypotenuse is 126225.

In other words, if you sum the squares of 88209 and its reverse, the result is a perfect square.

Podcast Episode 95: A New Day at Charleston

In 1862, slave Robert Smalls was working as a pilot aboard a Confederate transport ship in Charleston, S.C., when he siezed a unique chance to escape. In this week’s episode of the Futility Closet podcast we’ll follow his daring predawn journey, which rescued 17 people from slavery and changed the course of South Carolina history.

We’ll also reflect on justice for bears and puzzle over a hijacker’s surprising request.

Please consider becoming a patron of Futility Closet — on our Patreon page you can pledge any amount per episode, and all contributions are greatly appreciated. You can change or cancel your pledge at any time, and we’ve set up some rewards to help thank you for your support.

You can also make a one-time donation via the Donate button in the sidebar of the Futility Closet website.

Sources for our feature on Robert Smalls:

Andrew Billingsley, Yearning to Breathe Free: Robert Smalls of South Carolina and His Families, 2007.

Kitt Haley Alexander, Robert Smalls: First Black Civil War Hero, 2001.

Peggy Cooper Davis, “Introducing Robert Smalls,” Fordham Law Review 69:5 (April 2001), 1695.

“Robert Smalls,” American National Biography Online, accessed Feb. 14, 2016.

Henry Louis Gates Jr., “Which Slave Sailed Himself to Freedom?”, (accessed Feb. 14, 2016).

Micah White, “Black History Unsung Heroes: Robert Smalls,”, Feb 9, 2015.

“Smalls, Robert,” History, Art & Archives, U.S. House of Representatives (accessed Feb. 14, 2016).

Blain Roberts and Ethan J. Kytle, “Robert Smalls’s Great Escape,” New York Times, May 12, 2012.

Avis Thomas-Lester, “Civil War Hero Robert Smalls Seized the Opportunity to Be Free,” Washington Post, March 2, 2012.

Amy Geier Edgar, “Bill Would Honor Black Pioneer in Business, Politics,” Associated Press, March 26, 2004.

Listener mail:

Todd Wilkinson, “What Do You Do With a Bear That Kills a Person?”, National Geographic, Aug. 20, 2015.

Sarene Leeds, “‘Downton Abbey’ Recap: Season 6, Episode 5,” Wall Street Journal, Jan. 31, 2016.

This week’s lateral thinking puzzle was contributed by listener Rini Rikka.

You can listen using the player above, download this episode directly, or subscribe on iTunes or via the RSS feed at

Many thanks to Doug Ross for the music in this episode.

If you have any questions or comments you can reach us at Thanks for listening!

Another Day
Image: Wikimedia Commons

Flying alone over France in April 1917, German flying ace Ernst Udet engaged another lone pilot in aerial combat. The other pilot, a Frenchman, was exceptionally talented, anticipating all of Udet’s moves and reacting instantly. “Sometimes we pass so closely I can clearly recognize a narrow, pale face under the leather helmet,” Udet wrote later. “On the fuselage, between the wings, there is a word in black letters. As he passes me for the fifth time, so close that his propwash shakes me back and forth, I can make it out: ‘Vieux‘ it says there — vieux — the old one. That’s Guynemer’s sign.”

Guynemer was Georges Guynemer, France’s top fighter ace, who had brought down 30 Germans in fights like this. “Slowly I realize his superiority,” Udet wrote. “His aircraft is better, he can do more than I, but I continue to fight.” For a moment he managed to get Guynemer into his sights, but he found that his gun wouldn’t fire — it was blocked.

Udet tried to clear the stoppage by hand but failed. He considered diving away but knew that Guynemer would instantly shoot him down. They circled one another for another eight minutes as Udet sought to evade the Frenchman’s guns. When Guynemer swooped overhead, Udet hammered the gun with his fists and then realized his mistake:

Guynemer has observed this from above, he must have seen it, and now he knows what gives with me. He knows I’m helpless prey.

Again he skims over me, almost on his back. Then it happens: he sticks out his hand and waves to me, waves lightly, and dives to the west in the direction of his lines.

I fly home. I’m numb.

“There are people who claim Guynemer had a stoppage himself then,” Udet wrote in Ace of the Iron Cross. “Others claim he feared I might ram him in desperation. But I don’t believe any of them. I still believe to this day that a bit of chivalry from the past has continued to survive. For this reason I lay this belated wreath on Guynemer’s unknown grave.”


The world’s oldest operating roller coaster, Leap-the-Dips, in Altoona, Pa., was built in 1902. It’s 41 feet high and has an average speed of 10 mph.

New Jersey’s Kingda Ka, below, opened a century later. It’s 456 feet high and accelerates to 128 mph in 3.5 seconds.

What’s next?

In a Word

n. the act of restoring or repairing

adj. given to thieving, thievish

In 1996, workers demolishing the old Apollo Theater on West 42nd Street in New York City discovered a hidden cache of discarded wallets. Apparently a thief had preyed on theatergoers there 40 years earlier, stealing wallets and pocketbooks, removing the cash and valuables, and dropping the rest into an airshaft.

“The farther back I crawled, the older they got, from the 1960s to the 1950s,” foreman Bill Barron told the New York Times.

The finds included a weekly paycheck stub for $226.30, a telephone bill for $7.24, faded photographs, and identification papers of the victims, few of whom were still living.

“The Times Square of the late 1950s and early 1960s was the capital of pickpocketing,” said social historian Luc Sante. “It was simply a more trusting era.”

Steps Ahead

One early locomotive had legs. Scottish inventor William Brunton devised the “Mechanical Traveller” in 1813, giving it feet to grip the track on steep grades. It could creep forward at about 3 mph.

Popularly known as the “Grasshopper,” it hauled coal for about two years at the Newbottle Colliery until it ended its career with the first recorded railway disaster, a boiler explosion that killed 16 spectators. Brunton abandoned the project.

A Prime Number Generator

Take the first n prime numbers, 2, 3, 5, …, pn, and divide them into two groups in any way whatever. Find the product of the numbers in each group, and call these A and B. (If one of the groups is empty, assign it the product 1.) No matter how the numbers are grouped, A+B and \left |A-B  \right | will always turn out to be prime numbers, provided only that they’re less than p_{n+1}^{2} (and greater than 1, of course). For example, here’s what we get for (2, 3, 5) (where p_{n+1}^{2} = 72 = 49):

2 × 3 + 5 = 11
2 × 5 + 3 = 13
2 × 5 – 3 = 7
3 × 5 + 2 = 17
3 × 5 – 2 = 13
2 × 3 × 5 + 1 = 31
2 × 3 × 5 – 1 = 29

In More Mathematical Morsels (1991), Ross Honsberger writes, “For me, the fascination with this procedure seems to lie to a considerable extent in the amusement of watching it actually turn out prime numbers; I’m sure I only half believed it would work until I had seen it performed a few times.”

It makes sense if you think about it. Each of the first n prime numbers will divide either A or B but not the other, so it will fail to divide either A+B or \left |A-B  \right |. That means that any prime divisor of A+B or \left |A-B  \right | must be at least as big as p_{n+1}, and if there were more than one of them, the number would amount to at least p_{n+1}^{2}, putting it outside the limit. So for A+B or \left |A-B  \right | between 1 and p_{n+1}^{2}, it must itself be a prime number p such that pn+1p < p_{n+1}^{2}.


Charlie Chaplin demanded 342 takes for one three-minute scene in City Lights. Actress Virginia Cherrill played a blind flower girl who mistakes Chaplin for a wealthy man. Her only line was “Flower, sir?”

Chaplin later called Cherrill an “amateur”; he’d hired her as the love interest without even talking to her. Asked why so many takes were necessary, he said, “She’d be doing something which wasn’t right. Lines. A line. A contour hurts me if it’s not right. … I’d know in a minute when she wasn’t there, when she’d be searching, or looking up just too much or too soon … Or she waited a second. I’d know in a minute.”

But it’s also true that Chaplin often worked out a scene on the set, rather than relying on a finished script. “Chaplin rehearsed on film — he’d try out an idea and do it over and over again,” film historian Hooman Mehran, who narrates the segment above, told CNN. “And since he was the director, he couldn’t see his performance, so he had to record it.”