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)
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.”
Some of these numbers are palindromes, numbers that read the same backward and forward. A few examples are 55, 66, 171, 595, and 666. In 1973, Charles Trigg found that of the triangular numbers less than 151340, 27 are palindromes.
But interestingly, every string of 1s:
1
11
111
1111
11111
…
… is a palindromic triangular number in base nine. For example:
The pattern continues — all these numbers are triangular.
02/12/2017 UPDATE: Reader Jacob Bandes-Storch sent a visual proof:
“Given a number n in base 9, if we tack a 1 on the right, the resulting number is 9*n + 1. (By shifting over one place to the left, each digit becomes nine times its original value, and then we add 1 in the ones place.) So given a triangular number, there’s probably a way of sticking together 9 copies of it with a single additional unit to form a new triangle. Sure enough:”
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 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.
A harmonic progression is a progression formed by taking the reciprocals of an arithmetic progression (so an example is 1/1, 1/2, 1/3, 1/4 …). When tutoring mathematics at Oxford, Charles Dodgson had a favorite example to illustrate this:
According to him, it is (or was) the rule at Christ Church that, if an undergraduate is absent for a night during term-time without leave, he is for the first offence sent down for a term; if he commits the offence a second time, he is sent down for two terms; if a third time, Christ Church knows him no more. This last calamity Dodgson designated as ‘infinite.’ Here, then, the three degrees of punishment may be reckoned as 1, 2, infinity. These three figures represent three terms in an ascending series of Harmonic Progression, being the counterparts of 1, 1/2, 0, which are three terms in a descending Arithmetical Progression.
— Lionel A. Tollemache, “Reminiscences of ‘Lewis Carroll,'” Literature, Feb. 5, 1898
Dentist William T.G. Morton was the first to use sulfuric ether as an anaesthetic, but he’d learned about this property at the chemistry lectures of Charles T. Jackson.
Which of them deserved a monument? Oliver Wendell Holmes suggested setting up statues of both men on the same pedestal, with the inscription:
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.
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.”
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.”
