Bertrand’s Paradox

We ask for the probability that a number, integer or fractional, commensurable or incommensurable, randomly chosen between 0 and 100, is greater than 50. The answer seems evident: the number of favourable cases is half the number of possible cases. The probability is 1/2.

Instead of the number, however, we can choose its square. If the number is between 50 and 100, its square will be between 2,500 and 10,000.

The probability that a randomly chosen number between 0 and 10,000 is greater than 2,500 seems evident: the number of favourable cases is three quarters of the number of possible cases. The probability is 3/4.

The two problems are identical. Why are the two answers different?

— Joseph Bertrand, Calcul des probabilités, 1889 (translation by Sorin Bangu)

Through the Looking-Glass

In 2015, to celebrate the 150th anniversary of the publication of Alice’s Adventures in Wonderland, master sculptor Karen Mortillaro created 12 new sculptures, one for each chapter in Lewis Carroll’s masterpiece. Each takes the form of a table topped with an S-cylindrical mirror, with a bronze sculpture on either side. The sculpture that stands before the mirror is anamorphic, so that the curved mirror’s reflection “undistorts” it, giving it meaning:

“The S-cylindrical mirror is ideal for this project because it allows for the figures on one side of the mirror to be sculpted realistically, while those on the opposite side of the mirror are distorted and unrecognizable,” Mortillaro writes. “The mirror is symbolic of the parallel worlds that Alice might have experienced in her dream state; the world of reality is on one side of the mirror; and the world of illusion is on the mirror’s opposite side.”

Mortillaro’s article appears in the September 2015 issue of Recreational Mathematics Magazine.

R.I.P. Raymond Smullyan, 1919–2017

Philosopher and logician Raymond Smullyan passed away on Monday. He was 97.

From my notes, here’s a paradox he offered at a Copenhagen self-reference conference in 2002:

Have you heard of the LAA computing company? Do you know what LAA stands for? It stands for ‘lacking an acronym.’

Actually, the above acronym is not paradoxical; it is simply false. I thought of the following variant which is paradoxical — it is the LACA company. Here LACA stands for ‘lacking a correct acronym.’ Assuming that the company has no other acronym, that acronym is easily seen to be true if and only if it is false.


In Pascal’s triangle, above, the number in each cell is the sum of the two immediately above it.

If you “tilt” the triangle so that each row starts one column to the right of its predecessor, then the column totals produce the Fibonacci sequence:

pascal triangle fibonacci numbers

That’s from Thomas Koshy’s Triangular Arrays With Applications, 2011.

Bonus: Displace the rows still further and they’ll identify prime numbers.

Podcast Episode 140: Ramanujan

In 1913, English mathematician G.H. Hardy received a package from an unknown accounting clerk in India, with nine pages of mathematical results that he found “scarcely possible to believe.” In this week’s episode of the Futility Closet podcast, we’ll follow the unlikely friendship that sprang up between Hardy and Srinivasa Ramanujan, whom Hardy called “the most romantic figure in the recent history of mathematics.”

We’ll also probe Carson McCullers’ heart and puzzle over a well-proportioned amputee.


W.H. Hill’s signature was unchanged when inverted.

Room 308 of West Java’s Samudra Beach Hotel is reserved for the Indonesian goddess Nyai Loro Kidul.

Sources for our feature on Srinivasa Ramanujan:

Robert Kanigel, The Man Who Knew Infinity, 1991.

K. Srinivasa Rao, Srinivasa Ramanujan: A Mathematical Genius, 1998.

S.R. Ranganathan, Ramanujan: The Man and the Mathematician, 1967.

Bruce C. Berndt and Robert A. Rankin, Ramanujan: Letters and Commentary, 1991.

G.H. Hardy, “The Indian Mathematician Ramanujan,” American Mathematical Monthly 44:3 (March 1937), 137-155.

Gina Kolata, “Remembering a ‘Magical Genius,'” Science 236:4808 (June 19, 1987), 1519-1521.

E.H. Neville, “Srinivasa Ramanujan,” Nature 149:3776 (March 1942), 293.

Bruce C. Berndt, “Srinivasa Ramanujan,” American Scholar 58:2 (Spring 1989), 234-244.

B.M. Srikantia, “Srinivasa Ramanujan,” American Mathematical Monthly 35:5 (May 1928), 241-245.

S.G. Gindikin, “Ramanujan the Phenomenon,” Quantum 8:4 (March/April 1998), 4-9.

“Srinivasa Ramanujan” in Timothy Gowers, June Barrow-Green, and Imre Leader, eds., Princeton Companion to Mathematics, 2010.

“Srinivasa Aiyangar Ramanujan,” MacTutor History of Mathematics (accessed Jan. 22, 2017).

In the photo above, Ramanujan is at center and Hardy is at far right.

Listener mail:

“Myth Debunked: Audrey Hepburn Did Not Work for the Resistance” [in Dutch], Dutch Broadcast Foundation, Nov. 17, 2016.

“Audrey Hepburn’s Son Remembers Her Life,” Larry King Live, CNN, Dec. 24, 2003.

This week’s lateral thinking puzzle was contributed by listener Tyler Rousseau.

You can listen using the player above, download this episode directly, or subscribe on iTunes or Google Play Music 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 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!

Image: Wikimedia Commons

In 1965, as they were writing the first draft of 2001: A Space Odyssey, Stanley Kubrick showed Arthur C. Clarke a set of 12 plastic tiles. Each tile consisted of five squares joined along their edges. These are known as pentominoes, and a set of 12 includes every possible such configuration, if rotations and reflections aren’t considered distinct. The challenge, Kubrick explained, is to fit the 12 tiles together into a tidy rectangle. Because 12 five-square tiles cover 60 squares altogether, there are four possible rectangular solutions: 6 × 10, 5 × 12, 4 × 15, and 3 × 20. (A 2 × 30 rectangle would be too narrow to accommodate all the shapes.)

Clarke, who rarely played intellectual games, found that this challenge “can rather rapidly escalate — if you have that sort of mind — into a way of life.” He stole a set of tiles from his niece, spent hundreds of hours playing with it, and even worked the shapes into the design of a rug for his office. “That a jigsaw puzzle consisting of only 12 pieces cannot be quickly solved seems incredible, and no one will believe it until he has tried,” he wrote in the Sunday Telegraph Magazine. It took him a full month to arrange the 12 shapes into a 6 × 10 rectangle — a task that he was later abashed to learn can be done in 2339 different ways. There are 1010 solutions to the 5 × 12 rectangle and 368 solutions to the 4 × 15.
Image: Wikimedia Commons

But “The most interesting case, however, is that of the long, thin rectangle only 3 units wide and 20 long.” Clarke became fascinated with this challenge when Martin Gardner revealed that only two solutions exist. He offered 10 rupees to anyone who could find the solutions, and was delighted when a friend produced them, as he’d calculated that solving the problem by blind permutation would take more than 20 billion years.

Clarke even worked the 3 × 20 problem into his 1975 novel Imperial Earth. Challenged by his grandmother, the character Duncan struggles with the task and declares it impossible. “I’m glad you made the effort,” she says. “Generalizing — exploring every possibility — is what mathematics is all about. But you’re wrong. It can be done. There are just two solutions; and if you find one, you’ll also have the other.”

Can you find them?

Click for Answer

Art and Science

Alexander Fleming, the discoverer of pencillin, grew “germ paintings” of living bacteria on blotting paper. He made this 4-inch portrait, titled “Guardsman,” in 1933.

“If a paper disc is placed on the surface of an agar plate, the nutrient material diffuses through the paper sufficiently to maintain the growth of many microorganisms implanted on the surface of the paper,” he wrote. “At any stage, growth can be stopped by the introduction of formalin. Finally the paper disc, with the culture on its surface, can be removed, dried, and suitably mounted.”

Here’s a gallery. “Even in Fleming’s time this technique failed to receive much attention or approval. Apparently he prepared a small exhibit of bacterial art for a royal visit to St Mary’s by Queen Mary. The Queen was ‘not amused and hurried past it’ even though it included a patriotic rendition of the Union Jack in bacteria.”

Area Magic Squares

On December 30 William Walkington sent this greeting to a circle of magic-square enthusiasts — it’s a traditional magic square (each row, column, and diagonal sums to 15), but the geometric area of each cell corresponds to its number.

He added, “The areas are approximate, and I don’t know if it is possible to obtain the correct areas with 2 vertically slanted straight lines through the square. Perhaps someone will be able to work this out in 2017?”

It’s only January 19, and the answer is already yes — Walter Trump has produced a “third-order linear area magic square” using the numbers 5-13:

There are many further developments, which have opened new questions and challenges, as these discoveries tend to do — see William’s blog post for more information.

(Thanks, William.)

Watching the Detectives

Police exist, and sometimes they scrutinize other members of the constabulary. We might say Police police police. If the observed officers are already being observed by a third set of officers, then we could say Police police police police police, that is, “Police observe police [whom] police police.”

The trouble is that if you say this sentence, “Police police police police police,” to an innocent friend, she might take you to mean “Police [whom] police police … police police.” Police police police police police has one verb, police, and two noun phrases, Police and police police police, and without some guidance there’s no way to tell which noun phrase is intended to begin and which to end the sentence.

It gets worse. Suppose we add two more polices: Police police police police police police police. Now do we mean “Police [whom] police observe observe police [whom] police observe”? Or “Police observe police [whom] police whom police observe observe”? Or something else again?

In general, McGill University mathematician Joachim Lambek finds that if police is repeated 2n + 1 times (n ≥ 1), then the numbers of ways in which the sentence can be parsed is  \frac{1}{\left ( n + 1 \right )}\binom{2n}{n} , the (n + 1)st Catalan number.

Buffalo have their own troubles.

(J. Lambek, “Counting Ambiguous Meanings,” Mathematical Intelligencer 30:2 [March 2008], 4.)