Best Laid Plans

Suppose you hire two proofreaders to go through the same manuscript independently. The first reports A mistakes, the second reports B mistakes, and C mistakes are reported by both. How can you estimate how many errors remain undiscovered?

Let M be the total number of mistakes in the manuscript. Then the number undiscovered by the two proofreaders is M – (A + BC). Let p and q be the probabilities that the first and second proofreaders, respectively, notice any given mistake. Then ApM and BqM. And because they work independently, the chance that they both find a given mistake is CpqM.

But now

\displaystyle M = \frac{pM \times qM}{pqM} \approx \frac{AB}{C},

and the number of misprints that remain unnoticed is just

\displaystyle M - (A + B - C) \approx \frac{AB}{C} - (A + B - C) = \frac{(A-C)(B-C)}{C}.

This means that as long as the proofreaders work independently, you can estimate the number of errors they’ve overlooked without even knowing how skillful they are. If they find a large number of mistakes in common but relatively few independently, then the manuscript is probably relatively clean. But if they generate large independent lists of errors with few in common, there are probably many mistakes remaining to be found (which matches our intuition).

(George Pólya, “Probabilities in Proofreading,” American Mathematical Monthly 83:1 [January 1976], 42.)

Black and White

mate on the move

White to play and mate on his first move.

(Don’t spend too much time on this — it’s a bit of a trick.)

Click for Answer

In a Word

adj. pertaining to the big toe

In 1856, 10-year-old Ellen Terry was just about to give Puck’s final speech in A Midsummer Night’s Dream when a stagehand closed a trapdoor on her foot, breaking her toe. She screamed, but manager Ellen Kean offered to double her salary if she finished the play. So, supported by Kean on one side and her sister Kate on the other, she delivered the following soliloquy:

If we shadows have offended (Oh, Katie, Katie!)
Think but this, and all is mended, (Oh, my toe!)
That you have but slumbered here,
While these visions did appear. (I can’t, I can’t!)
And this weak and idle theme,
No more yielding but a dream, (Oh, dear! oh, dear!)
Gentles, do not reprehend; (A big sob)
If you pardon, we will mend. (Oh, Mrs. Kean!)

“How I got through it, I don’t know!” she wrote in her 1908 memoir. “But my salary was doubled — it had been fifteen shillings, and it was raised to thirty — and Mr. Skey, President of Bartholomew’s Hospital, who chanced to be in a stall that very evening, came round behind the scenes and put my toe right. He remained my friend for life.”


FC book covers

Just a reminder — Futility Closet books make great gifts for people who are impossible to buy gifts for. Both contain hundreds of hand-picked favorites from our 11-year archive of curiosities. Some sample entries from the first book’s index:

gaiety, of bespectacled horses, 35
hallucination, useful in enlivening Bristol, 94
Nazis, poorly informed regarding Irish railway schedules, 36
chili sauce, and estate planning, 25
softball, at North Pole, 67
whinging, vaudevillian, placated, 86
corned beef sandwiches, weightless, 170
Nagel, Conrad, induced to reflect wearily upon his romantic choices, 116

And from the second:

communists, driving habits of, 73
trombones, and hair loss, 139
prayers, Canadian, private, whether, 179
James, Henry, not a refulgent fireball of novelistic rigor, 174
plunges, earthward, righted dangerously, 111
whales, exasperated, 193
Carroll, Lewis, beset by asparagus, 174
Story magazine, hard up for Ws, 19

Both also feature the curious words, odd inventions, intriguing quotations, and challenging puzzles that are regular features on the site. Boing Boing founder Mark Frauenfelder says, “Futility Closet delivers concentrated doses of weird, wonderful, brain-stimulating ideas and anecdotes, curated mainly from forgotten old books. I’m hooked — there’s nothing quite like it!”

Buy them now on Amazon!

Breaking the News

In the December 2012 issue of 1 Across magazine, longtime crossword composer John Graham included a special instruction above one of his puzzles:

“I have 18dn of the 19; no 27, just 13 15; no 2 or 6 or 1dn 26 yet — plenty of 10, though I wouldn’t have chosen the timing.”

Solvers discovered that 18 down was CANCER and 19 across was OESOPHAGUS. The full message read:

“I have CANCER of the OESOPHAGUS; no CHEMOTHERAPY, just PALLIATIVE CARE; no NARCOTIC or STENT or MACMILLAN NURSE yet — plenty of MERRIMENT, though I wouldn’t have chosen the timing.”

The puzzle was reprinted as cryptic crossword No. 25,842 in the Guardian the following month.

“It seemed the natural thing to do somehow,” Graham said. “It just seemed right.” He died in November 2013, and the Guardian published a tribute crossword to remember him.

(Thanks, Anthony.)

A Story Machine

Here’s a curious invention from 1916, in the early days of motion pictures: It’s a machine designed to suggest plot ideas by randomly juxtaposing ideas. Words, pictures, and even bars of music are printed on paper rollers, and the writer turns these to present six elements that form the basis of a story.

In the example above, the machine presents the words aged, aviator, bribes, cannibal, carousal, and escape. “These particular words readily suggest, for instance, that an aged aviator after flying through the air on a long trip, lands finally on a desolate island where he is met by a cannibal, whom he is forced to bribe to secure his safety. After an interim which is full of possibilities as a basis of a story, a carousal ensues following which the aviator escapes.”

Inventor Arthur Blanchard says that this technique can be used to inspire any fictional work, from a cartoon to a song, but he patented it specifically as a “movie writer.” Whether it inspired any movies I don’t know.

Three Sides

equilateral areas

If an equilateral triangle is inscribed in, and has a common vertex with, a rectangle, as shown above, then areas A + B = C.

If a triangle with angles α, β, γ is inscribed in, and has a common vertex with, a rectangle, as shown below, and if the right triangles opposite α, β, γ have areas A, B, C, respectively, then A cot α + B cot β = C cot γ.


Somewhat related: A Curious Equality.

(Tom M. Apostol and Mamikon Mnatsakanian, “Triangles in Rectangles,” Math Horizons 5:3 [February 1998], 29-31.)

A Last Look

On Sept. 26, 1901, 13-year-old Fleetwood Lindley was attending school in Springfield, Ill., when his teacher handed him a note: His father wanted him urgently. He rode his bicycle to the Oak Ridge cemetery two miles out of town and found his father, Joseph, in the memorial hall of Abraham Lincoln’s tomb. The assassinated president, now 36 years dead, was being transferred to a new resting place, and a small group of caretakers had decided to open his coffin to confirm his identity.

The casket had been laid across a pair of sawhorses. A pair of workmen used a blowtorch to unseal the lead panel that covered Lincoln’s upper body, and the small group peered in.

Afterward the coffin was lowered into a hole 10 feet deep, encased in a cage of steel bars, and buried under tons of concrete. Over the years, as the other witnesses passed away, Lindley became the last living person to have looked on Lincoln’s body.

“His face was chalky white,” he remembered for a Life reporter in 1963, three days before his own death. “His clothes were mildewed. And I was allowed to hold one of the leather straps as we lowered the casket for the concrete to be poured.”

“I was not scared at the time, but I slept with Lincoln for the next six months.”

Podcast Episode 83: Nuclear Close Calls

In 1983, Soviet satellites reported that the United States had launched a nuclear missile toward Moscow, and one officer had only minutes to decide whether to initiate a counterstrike. In today’s show we’ll learn about some nuclear near misses from the Cold War that came to light only decades after they occurred.

We’ll also hear listeners’ input about crescent moons and newcomers to India, and puzzle over the fatal consequences of a man’s departure from his job.

Sources for our feature on Stanislav Petrov and Vasili Arkhipov:

Pavel Aksenov, “Stanislav Petrov: The Man Who May Have Saved the World,” BBC, Sept. 26, 2013.

Lynn Berry, “Russian Who ‘Saved the World’ Recalls His Decision as 50/50,” Associated Press, Sept. 17, 2015.

“Soviet Officer Honored for Averting Nuclear War,” Toledo Blade, May 22, 2004.

Mark McDonald, “Cold War, Cool Head,” Milwaukee Journal Sentinel, Dec. 26, 2004.

Ben Hoyle, “The Russian Who Saved the World,” Southland Times, May 22, 2015, 7.

Glen Pedersen, “Stanislav Petrov, World Hero,” Fellowship, July/August 2005, 9.

“JFK Tried to Drive Wedge Between Cubans, Soviets,” Toledo Blade, Oct. 13, 2002.

“Papers: Annihilation Narrowly Averted,” Lawrence [Kan.] Journal-World, Oct. 12, 2002.

“Revealed: Soviet Sub Almost Attacked in ’62,” Peace Magazine, January-March 2003, 31.

Listener mail:

The Museum of London’s exhibition The Crime Museum Uncovered runs through April 10, 2016.

Wordnik defines griffinism as “In India and the East, the state or character of a griffin or new-comer.”

This week’s lateral thinking puzzle was contributed by listener Andrew H., 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

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.

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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!