The square root of 2 is 1.41421356237 … Multiply this successively by 1, by 2, by 3, and so on, writing down each result without its fractional part:

two-timing 1

Beneath this, make a list of the numbers that are missing from the first sequence:

two-timing 2

The difference between the upper and lower numbers in these pairs is 2, 4, 6, 8 …

From Roland Sprague, Recreations in Mathematics, 1963.

Kirkman Schoolgirls Problem

“Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange them daily, so that no two shall walk twice abreast.”

The Rev. Thomas Penyngton Kirkman posed this query innocently in The Lady’s and Gentleman’s Diary in 1850. It’s trickier than it looks — though seven solutions are possible, they’re difficult to discover by trial and error. Here’s one:

kirkman schoolgirl solution

Each letter appears once in each row, but no two letters share a cell more than once. The problem’s simplicity made it popular among Victorian amateur mathematicians, and Kirkman later complained that it had eclipsed his more serious work — though he took pains to dispute James Joseph Sylvester’s claim to have invented the problem himself.

Do such problems generally have solutions? Surprisingly, the answer remained unknown until just last January, when Oxford mathematician Peter Keevash showed that the answer is yes if certain basic requirements are met. Keevash’s result was “a bit of an earthquake as far as design theory is concerned,” said Cambridge mathematician Timothy Gowers.

Now that the schoolgirl problem is under control, a related challenge has taken its place. If 20 golfers want to arrange themselves into different foursomes on five successive days, is it possible to plan the groups so that each golfer plays no more than once with any other golfer? Formally the “social golfer problem” remains unsolved — though tournament organizers work out individual solutions every day.

(Thanks, Martha.)

Law and Punctuation

In Ward v. Ward, a North Carolina court’s entire finding rested on the presence of a single comma. The will of Alvin T. Ward read:

My Trustee is directed to pay such amounts of and from the income generated by said trust, and from the principal of said trust if he deems same to be advisable, to, for, or on account of my said wife in quarterly installments or more frequently if he deems advisable and if practicable.

Is the trustee required to pay the income to Ward’s wife, or can he use his discretion? The income payments are required, ruled the court: The comma after “said trust” shows that only the distribution of the principal is left to the trustee’s judgment.

In Henderson v. State, Jacob Henderson’s 1984 burglary conviction in Mississippi was reversed in part because of a misplaced period:

The Grand Jurors for the State of Mississippi, … upon their oaths present: That Jacob Henderson … on the 15th day of May, A.D., 1982.

This is a “non-sentence,” noted the court. “The unmistakable period after 1982 is used by astute defense counsel to nail down the point — that the indictment fails to charge that Jacob Henderson did anything on May 15, 1982.” (This whole indictment is a notorious trainwreck.)

And in People v. Vasquez, a New York court disregarded an affidavit and dismissed a complaint because a misplaced comma made it unclear whether a key affidavit was hearsay:

“It may be that the confusion [about the affidavit] arises from the typographical error of placing a comma before the expression ‘upon information and belief,'” wrote the court. “Had the comma not existed, the entire expression ‘and that the assertion upon information and belief’ would have referred back to the earlier mentioned accusatory instruments so as to render the affidavit non-hearsay.”

Hour of Babel

A problem from the second Balkan Mathematical Olympiad, 1985:

Of the 1985 people attending an international meeting, no one speaks more than five languages, and in any subset of three attendees, at least two speak a common language. Prove that some language is spoken by at least 200 of the attendees.

Click for Answer

Podcast Episode 64: Murder at the Priory

In 1876 London was riveted by the dramatic poisoning of a young barrister and the sordid revelations that emerged about his household. In today’s show we’ll review the baffling case of Charles Bravo’s murder, which Agatha Christie called “one of the most mysterious poisoning cases ever recorded.”

We’ll also get an update on career possibilities for garden hermits and puzzle over how the police know that a shooting death is not a homicide.

Many thanks to Ronald Hackston for his evocative photo of The Priory, Balham, the site of Charles Bravo’s unsolved 1876 poisoning.

Sources for that feature:

James Ruddick, Death at the Priory: Sex, Love, and Murder in Victorian England, 2001.

Chirag Trivedi, “Victorian Whodunnit Solved,” BBC, Jan. 13, 2003 (accessed June 28, 2015).

“The Bravo Inquiry” and “The Theory of Suicide in the Bravo Case,” Medical Times and Gazette, Aug. 19, 1876.

Joyce Emmerson Muddock, Pages From an Adventurous Life, 1907.

Listener mail:

Amanda Williams, “Wanted: ‘Outgoing’ Hermit,” Daily Mail, May 5, 2014 (retrieved July 3, 2015).

Greater Manchester News, “Hermit Wanted for Historic Gardens,” July 3, 2009 (retrieved July 3, 2015).

“Hermit Wanted for ‘Ivory Tower’,” BBC, July 1, 2009 (retrieved July 3, 2015).

This week’s lateral thinking puzzle was contributed by listener Sam B., who sent this corroborating link (warning: this spoils the puzzle).

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

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“Happiness, whether consisting in pleasure or virtue, or both, is more often found with those who are highly cultivated in their minds and in their character, and have only a moderate share of external goods, than among those who possess external goods to a useless extent but are deficient in higher qualities.” — Aristotle

“Money only appeals to selfishness and always tempts its owners irresistibly to abuse it. Can anyone imagine Moses, Jesus, or Gandhi armed with the money-bags of Carnegie?” — Albert Einstein, The World As I See It, 1949

Where wealth and freedom reign, contentment fails,
And honour sinks where commerce long prevails.

— Oliver Goldsmith, The Traveller, 1764

“The essence of philosophy is that a man should so live that his happiness shall depend as little as possible on external things.” – Epictetus

“The most necessary disposition to relish pleasures is to know how to be without them.” — Marquise de Lambert, A Mother’s Advice to Her Son, 1726

“It is preoccupation with possessions, more than anything else, that prevents us from living freely and nobly.” — Bertrand Russell

“Who is rich? He that is content. Who is that? Nobody.” — Benjamin Franklin

Truth and Fiction

In Lillian Hellman’s 1973 memoir Pentimento, she describes a childhood friend whom she calls “Julia” who became active in the Austrian underground during World War II. The book was made into the Oscar-winning 1977 film Julia, starring Jane Fonda and Vanessa Redgrave.

But after the book appeared, readers noticed something peculiar. Julia strongly resembled a real person, Muriel Gardiner, a psychoanalyst. Both women were millionaires’ daughters who had attended Wellesley and Oxford, moved to Vienna to study with Freud, bore daughters, became socialists, and participated in anti-Fascist and anti-Nazi activities before the war. But where Gardiner sailed for the United States in 1939, Hellman’s Julia was tortured to death by Nazis. Hellman claimed that she flew the body home but had it cremated when she was unable to find Julia’s mother.

Despite all these similarities, Hellman insisted that Julia was a different person and said she had never heard of Gardiner. “She may have been the model for somebody else’s Julia,” she told the New York Times, “but she was certainly not the model for my Julia.” She said she refused to reveal her own Julia’s name for personal and legal reasons.

Gardiner wrote to Hellman in 1976, inquiring about all this, but never received a reply. She had kept silent about her activities for 40 years — but it’s notable that lawyer Wolf Schwabacher had socialized with Hellman in Europe while Gardiner was opposing Fascism in Vienna, and also shared a house with Gardiner after the war.

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