Close Quarters

Paulo Guarini di Forli offered this puzzle in 1512. On this tiny 3 × 3 board, which is the smallest number of moves in which the white knights can exchange places with the black ones?

Click for Answer

Podcast Episode 205: The White Mouse

In 1928 Nancy Wake ran away from her Australian home and into an unlikely destiny: She became a dynamo in the French resistance, helping more than a thousand people to flee the Germans and then organizing partisans to fight them directly. In this week’s episode of the Futility Closet podcast we’ll tell the story of the White Mouse, one of the bravest heroes of World War II.

We’ll also marvel at mailmen and puzzle over an expensive homework assignment.


The town of Agloe, New York, was invented by a pair of mapmakers.

F. Scott Fitzgerald’s This Side of Paradise contains two hidden poems.

Sources for our feature on Nancy Wake:

Peter FitzSimons, Nancy Wake, 2001.

Nancy Wake, The White Mouse, 1985.

Russell Braddon, The White Mouse, 1956.

“Dispatches,” World War II 26:4 (November/December 2011), 16.

“History in the Media,” History Today 55:4 (April 2005), 9.

“Sound Off,” Leatherneck 85:6 (June 2002), 2.

Adam Bernstein, “Nancy Wake, ‘White Mouse’ of World War II, Dies at 98,” Washington Post, Aug. 9, 2011.

Paul Vitello, “Nancy Wake, Proud Spy and Nazi Foe, Dies at 98,” New York Times, Aug. 13, 2011.

“Obituary: Nancy Wake,” Economist 400:8746 (Aug. 13, 2011), 82.

Chris Brice, “The Mouse That Roared,” [Adelaide] Advertiser, June 2, 2001.

Bruce Wilson, “Forever in Her Debt,” [Brisbane] Courier-Mail, Feb. 15, 2003, 34.

“War Heroine Nancy Wake Dies,” ABC Premium News, Aug. 8, 2011.

“Prince Helps Pauper Heroine,” [Adelaide] Advertiser, Feb. 11, 2003, 22.

“Australian ‘White Mouse’ Was a Guerrilla to Nazis Selling Her War Medals Did Not Endear Her to Countrymen, Though,” Christian Science Monitor, June 8, 1994.

Sandra Laville, “Penniless Resistance Hero Stays On … and On … at Hotel,” Vancouver Sun, Feb. 11, 2003, A16.

Red Harrison, “All Guts and Garters,” Weekend Australian, June 9, 2001.

Lydia Clifford, “Secrets and White Lies,” Daily Telegraph, June 1, 2001, 117.

Bruce Wilson, “Penniless Wake Is Also Priceless,” Daily Telegraph, Feb. 14, 2003, 23.

Nate Rawlings, “Nancy Wake,” Time 178:8 (Aug. 29, 2011), 20.

Roderick Bailey, “Wake, Nancy Grace Augusta,” Oxford Dictionary of National Biography, Jan. 8, 2015.

Listener mail:

A 1797 George III Cartwheel penny, a handgun, and a selection of pottery and pipes found on the Thames foreshore.

The Bozeman Daily Chronicle’s “Police Reports.”

The neural net that Dave Lawrence fed them through.

This week’s lateral thinking puzzle was contributed by listener Simone Hilliard, who sent this corroborating link (warning — this spoils the puzzle).

You can listen using the player above, download this episode directly, or subscribe on Apple Podcasts or Google Play Music or via the RSS feed at

Please consider becoming a patron of Futility Closet — you can choose the amount you want to pledge, and we’ve set up some rewards to help thank you for your support. You can also make a one-time donation on the Support Us page of the Futility Closet website.

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!

For What It’s Worth

In an effort to attract female worshippers and tourists, Taiwan’s Southwest Coast National Scenic Area erected the “High-Heel Wedding Church” in 2016 in Budai Township, Chiayi County. It’s 17 meters tall and contains about 320 pieces of blue-tinted glass; Guinness has certified it as the “world’s largest high-heel shoe-shaped structure.” The BBC reports:

The shoe was inspired by a local story. According to officials in the 1960s, a 24-year-old girl surnamed Wang from the impoverished region suffered from Blackfoot disease. Both of her legs had to be amputated, leading to the cancellation of her wedding. She remained unmarried and spent the rest of her life at a church. The high heel is intended to honour her memory.

The church will be used for weddings and pre-wedding photo shoots, not regular services. Pan Tsuei-ping, the administration’s recreation section manager, said, “In our planning, we want to make it a blissful, romantic avenue. … Every girl imagines how they will look like when they become the bride.” The building will contain “100 female-oriented features,” including “chairs for lovers, maple leaves, biscuits, and cakes,” all suited for romantic photographs.

But the project has also been criticized as sexist and patronizing. On the Chinese microblogging site Weibo, Jessie Chou wrote, “I wear flip flops anyway.”

The No-Three-in-Line Problem

In 1917 Henry Dudeney asked: What’s the maximum number of lattice points that can be placed on an n × n grid so that no three points are collinear?

The answer can’t be more than 2n, since if we place one point more than this, we’re forced to put three into the same row or column. (The 10 × 10 grid above contains 20 points.)

For a grid of each size up to 52 × 52, it’s possible to place 2n points without making a triple. For larger grids it’s conjectured that fewer than 2n points are possible, but today, more than a century after Dudeney posed the question, a final answer has yet to be found.

The Double Day

Lyndon Johnson averaged only 3 to 4 hours of sleep a night and worked most of the rest; his wife once said, “Lyndon acts as if there is never going to be a tomorrow.” He arranged his time in a curious pattern:

Johnson began every day with a bedroom conference at 6:30 a.m., then worked straight through until 2:00 p.m., when he had lunch, relaxed, sometimes with a swim, and took a quick nap. By 4:00 p.m. he was ready to go again. ‘It’s like starting a new day,’ Johnson observed, and he would then proceed to work straight through to one or two in the morning. This Johnsonian ‘double day’ amazed the press and exhausted and frustrated his over-worked aides. His assistant Jack Valenti opined that Johnson had ‘extra glands’ that gave him energy that ordinary men did not possess: ‘He goes to bed late, rises early, and the words I have never heard him say are “I’m tired.”‘

He once called a congressman at 3 a.m. to discuss a piece of pending legislation. When Johnson asked, “Were you asleep?” the congressman thought quickly and said, “No, Mr. President, I was just lying here hoping you’d call.”

(From Larry F. Vrzalik and Michael Minor, From the President’s Pen, 1991.)


jealousy glass

This is sneaky: Operagoers in the 18th century could spy on their neighbors using a “jealousy glass” — you’d appear to be watching the stage but a mirror would direct your view to the side, like a horizontal periscope. Marc Thomin, optician to the queen of France, wrote in 1749:

It is sufficient to turn this opening in the direction of whatever one wishes to observe and the curiosity is immediately satisfied. Its usefulness is confined to letting us see surreptitiously a person we seem not to be observing. This lorgnette may have been called a decorum glass because there is nothing more rude than to use an ordinary opera glass for looking at some one face to face.

Hanneke Grootenboer writes in Treasuring the Gaze, “Apparently, it was very convenient in allowing one to keep track of latecomers entering the opera without having to turn one’s head.”

(From J. William Rosenthal, From Spectacles and Other Vision Aids: A History and Guide to Collecting, 1996.)

Lost Voices

In 2009 three historians engaged forensic lip reader Jessica Rees to analyze silent film shot at the Battle of the Somme in 1916.

Soldiers of the Essex Regiment washing at a pool shout “Hi Mum!” and “Hello Mum, it’s me.” A soldier with a wounded foot repeats, “Jesus, Jesus, Jesus, Jesus.” And another soldier tells the crew, “Stop filming, this is awful.”

“What struck me the most was the optimism of the soldiers and their bravery,” Rees said. “They all seemed very positive, full of team spirit and jocular. Yet, as I was stunned to learn, many of them did not even survive the day of filming. I came away feeling a bit humble.”

The Nimm0 Property

In the 17th century the French mathematician Bernard Frénicle de Bessy described all 880 possible order-4 magic squares — that is, all the ways in which the numbers 1 to 16 can be arranged in a 4 × 4 array so that the long diagonals and all the rows and columns have the same sum.

These squares share a curious property: If we subtract 1 from each cell, to get a square of the numbers 0-15, then each of the rows and columns has a nim sum of 0. A nim sum is a binary sum in which 1 + 1 is evaluated as 0 rather than “0, carry 1.” For example, here’s one of Frénicle’s squares:

\displaystyle   \begin{matrix}  0 & 5 & 10 & 15\\   14 & 11 & 4 & 1\\   13 & 8 & 7 & 2\\   3 & 6 & 9 & 12  \end{matrix}

Translating each of these numbers into binary we get

\displaystyle   \begin{bmatrix}  0000 & 0101 & 1010 & 1111\\   1110 & 1011 & 0100 & 0001\\   1101 & 1000 & 0111 & 0010\\   0011 & 0110 & 1001 & 1100  \end{bmatrix}

And the binary sums of the four rows, evaluated without carry, are

0000 + 0101 + 1010 + 1111 = 0000
1110 + 1011 + 0100 + 0001 = 0000
1101 + 1000 + 0111 + 0010 = 0000
0011 + 0110 + 1001 + 1100 = 0000

The same is true of the columns. (The diagonals won’t necessarily sum to zero, but they will equal one another. And note that the property described above won’t necessarily work in a “submagic” square in which the diagonals don’t add to the magic constant … but it does work in all 880 of Frénicle’s “true” 4 × 4 squares.)

(John Conway, Simon Norton, and Alex Ryba, “Frenicle’s 880 Magic Squares,” in Jennifer Beineke and Jason Rosenhouse, eds., The Mathematics of Various Entertaining Subjects, Vol. 2, 2017.)

Needs Analysis

I do not believe in freedom of will. Schopenhauer’s words, ‘Man can indeed do what he wants, but he cannot want what he wants,’ accompany me in all life situations and console me in my dealings with people, even those that are really painful to me. This recognition of the unfreedom of the will protects me from taking myself and my fellow men too seriously as acting and judging individuals and losing good humor.

— Albert Einstein, Mein Glaubensbekenntnis, August 1932