The German Tank Problem,_Panzer_V_%22Panther%22.jpg

During World War II, as they mulled whether to attempt an invasion of the continent, the Allies needed to estimate the number of tanks Germany was producing. They asked their intelligence services to guess the number by spying on German factories and counting tanks on the battlefield, but these efforts produced contradictory estimates. Finally they resorted to statistical analysis.

They did this by studying the serial numbers on captured and destroyed German tanks. Suppose German tanks are numbered sequentially 1, 2, 3, …, B, where B is the total number of tanks that we seek to know. And suppose that we have five captured tanks whose serial numbers are 21, 35, 42, 60, and 89. It turns out that

\displaystyle B = \frac{(N+1)M}{N} - 1,

where N is the sample size (here, 5) and M is the highest sampled number (here, 89). In this example, the formula tells us that B = 105.8, so we’d estimate that 106 tanks had been produced at that time.

In the event, Allied statisticians reportedly estimated that the Germans had produced 246 tanks per month between June 1940 and September 1942. Intelligence estimates had put the total at about 1,400. When the Allies captured German production records after the war, they found that they had produced 245 tanks per month during those three years, almost precisely what the statisticians had predicted, and less than 20 percent of the intelligence estimate.

(Thanks, Ryan.)

Good Fortune

In 1777, in conversation with diplomat Arthur Lee, Benjamin Franklin reflected on the “miracle” of the American Revolution:

To comprehend it we must view a whole people for some months without any laws or government at all. In this state their civil governments were to be formed, an army and navy were to be provided by those who had neither a ship of war, a company of soldiers, nor magazines, arms, artillery or ammunition. Alliances were to be formed, for they had none. All this was to be done, not at leisure nor in a time of tranquillity and communication with other nations, but in the face of a most formidable invasion, by the most powerful nation, fully provided with armies, fleets, and all the instruments of destruction, powerfully allied and aided, the commerce with other nations in a great measure stopped up, and every power from whom they could expect to procure arms, artillery, and ammunition, having by the influence of their enemies forbade their subjects to supply them on any pretence whatever. Nor was this all; they had internal opposition to encounter, which alone would seem sufficient to have frustrated all their efforts. … It was, however, formed and established in despite of all these obstacles, with an expedition, energy, wisdom, and success of which most certainly the whole history of human affairs has not, hitherto, given an example.

“He told me the manner in which the whole of this business had been conducted, was such a miracle in human affairs, that if he had not been in the midst of it, and seen all the movements, he could not have comprehended how it was effected.”

(From Lee’s journal, Oct. 25, 1777.)

Different Strokes

In 1964, Swedish journalist Åke Axelsson paid a zookeeper to give a brush and paint to a 4-year-old chimpanzee named Peter. Then he chose the best of Peter’s paintings and exhibited them at the Gallerie Christinae in Göteborg, saying they were the work of a previously unknown French artist named Pierre Brassau.

Critic Rolf Anderberg of the Göteborgs-Posten wrote, “Brassau paints with powerful strokes, but also with clear determination. His brush strokes twist with furious fastidiousness. Pierre is an artist who performs with the delicacy of a ballet dancer.”

After Axelsson revealed the hoax, Anderberg maintained that Peter’s work was “still the best painting in the exhibition.”


In one oft-repeated anecdote from the memoirs of Melville Stone, publisher of the Chicago Daily News in the 1870s, the News suspected that the Chicago Post and Mail, published by the McMullen brothers, was pirating its stories. The News retaliated by printing an account of a famine in Serbia, in which the local mayor was quoted as saying (ostensibly in Serbian) ‘Er us siht la Etsll iws nel lum cmeht.’ When the afternoon edition of the Post and Mail duly reproduced the quote, Stone ran to all the other Chicago papers to reveal the hoax: read backward, the supposed quote said ‘The McMullens will steal this sure.’ According to Stone, the Post and Mail never recovered from the embarrassment, and the Daily News was able to buy it for a pittance less than two years later.

— Stuart Banner, American Property: A History of How, Why, and What We Own, 2011

(Thanks, Keith.)

Podcast Episode 97: The Villisca Ax Murders
Image: Flickr

Early one morning in 1912, the residents of Villisca, Iowa, discovered a horrible scene: An entire family had been brutally murdered in their sleep. In this week’s episode of the Futility Closet podcast we’ll describe the gruesome crime, which has baffled investigators for a hundred years.

We’ll also follow the further adventures of German sea ace Felix von Luckner and puzzle over some fickle bodyguards.

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 the Villisca ax murders:

Roy Marshall, Villisca, 2003.

“Suspect Is Held for Ax Murders,” [Spokane, Wash.] Spokesman-Review, May 15, 1917.

“Says He Killed Eight at God’s Command,” New York Times, Sept. 2, 1917.

“Tells of Killing Six With an Axe in 1912,” Associated Press, March 29, 1931.

“Iowa Town Marks 90th Anniversary of Unsolved Ax Murders,” Associated Press, June 9, 2002.

“Infamous Villisca Ax Donated to Villisca Historical Society,” Spencer [Iowa] Daily Reporter, Oct. 31, 2006.

Listener Rini Rikka writes, “Doch is very hard to comprehend for someone who is just starting to learn German. Besides the main usage as a short answer, it has lots of other meanings that help shorten the speech a bit. Unfortunately for the non-natives, those other meanings cannot always be translated with the same word, but with some practice you’ll get the feeling where and how to use it. If you’d like to read about it, here’s a good explanation of the word in English.”

This week’s lateral thinking puzzle was contributed by listener David White, who sent these corroborating links (warning: these spoil the puzzle).

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!

Dinner Wit

It’s said that when Frederick the Great hosted Voltaire at Sanssouci Palace, he sent him this puzzling note:
Image: Wikimedia Commons

It’s a rebus in French: deux mains sous Pé à cent sous scie? (“two hands under ‘p’ at hundred under saw”) means demain souper à Sanssouci? (“supper tomorrow at Sanssouci?”).

Voltaire replied “Ga!”: Gé grand, A petit! (“big ‘G’, small ‘a’!”) means j’ai grand appétit!, or “I am very hungry!”

Town and Country

More chess masters reside in New York City than in the rest of the United States combined. We’re planning a chess tournament that all American masters are expected to attend, and we want to minimize the total intercity traveling done by the players. The New York players argue that, by this criterion, the tournament should be held in their city. The West Coast players argue that a city should be chosen near the center of the gravity of the players. Where should we hold the tournament?

Click for Answer


first ladies

There are no known pictures of two American presidents’ wives: Martha Jefferson and Margaret Taylor.

We have one silhouette (left) of Jefferson, who was a little over 5 feet tall and had auburn hair and hazel eyes.

And one 1903 book contains a suggested likeness of Taylor (right), who was described during her life as “a fat, motherly looking woman,” “countenance rather stern but it may be the consequence of military association.”

But no portrait of either woman is known to exist. Some artists have attempted renderings based on pictures of their daughters, whom they were said to resemble, but that’s the best we can do.

Living Large

In his 1984 book Scaling, Duke University physiologist Knut Schmidt-Nielsen points out a pleasing coincidence:

A 30-gram mouse that breathes at a rate of 150 times per minute will breathe about 200 million times during its 3-year life; a 5-ton elephant that breathes at the rate of 6 times per minute will take approximately the same number of breaths during its 40-year lifespan. The heart of the mouse, ticking away at 600 beats per minute, will give the mouse some 800 million heartbeats in its lifetime. The elephant, with its heart beating 30 times per minute, is awarded the same number of heartbeats during its life.

In fact, most mammals have roughly the same number of heartbeats per lifetime, about 109. Small mammals have high metabolic rates and short lives; large ones have low rates and long lives. Humans are lucky: “We live several times as long as our body size suggests we should.”

In a Word

adj. inventive

adj. pertaining to flying

n. a burnt smell

Newsreel men recently witnessed an unscheduled drama as flames ended the attempt of Constantinos Vlachos, co-inventor of one of the strangest of flying craft, to win government aid for its development. He had planned an ascent from the lawn of the Congressional Library at Washington, D.C., to demonstrate his ‘triphibian,’ which he claimed could navigate in the air, on land, or in the water. Hardly had he started the motor when fire enveloped the machine. Spectators dashed to his aid and dragged him, severely burned, from the blazing wreck.

Popular Science, January 1936

(Thanks, Tucker.)