Science & Math

Fixing Dates

In 1899, British statistician Moses B. Cotsworth noted that recordkeeping could be greatly simplified if each month contained a uniform number of whole weeks. He proposed an “international fixed calendar” containing 13 months of 28 days each:

international fixed calendar

This makes everything easier. The 26th of every month falls reliably on a Thursday, for example, and statistical comparisons between months are made more accurate, as each month contains four tidy weeks with four weekends. (Unfortunately for the superstitious, every one of the 13 months contains a Friday the 13th.) A new month, called Sol, would be wedged between June and July, and an extra day, “Year Day,” would be added at the end of the year, but it would be independent of any month (as would Leap Day).

In 1922 the League of Nations chose Cotsworth’s plan as the most promising of 130 proposed calendar reforms, but the public, as always, resisted the unfamiliar, and by 1937 the International Fixed Calendar League had closed its doors. It left one curious legacy, though: George Eastman, the founder of Eastman Kodak, was so pleased with Cotsworth’s scheme that he adopted it as his company’s official calendar — and it remained so until 1989.


  • Babe Ruth struck out 1,330 times.
  • EMBARGO spelled backward is O GRAB ME.
  • The numbers on a roulette wheel add to 666.
  • The fourth root of 2143/22 is nearly pi (3.14159265258).
  • “A prosperous fool is a grievous burden.” — Aeschylus

Six countries have names that begin with the letter K, and each has a different vowel as the second letter: Kazakhstan, Kenya, Kiribati, Kosovo, Kuwait, Kyrgyzstan.

(Thanks, Danny.)

The Ulysses Contract

In 1982, 24-year-old schizophrenic patient J.S. faced a difficult decision: The neuroleptic drug Prolixin relieved his psychotic symptoms, but it produced tardive dyskinesia, a progressive disorder that caused uncontrollable movements of his legs, arms, and tongue.

His therapist learned of an experimental program that might reduce this side effect, and J.S. signed consent forms to enter treatment. But the first step was to stop all medications, and without the Prolixin he descended again into psychosis and refused the experimental medication.

This produces an impossible dilemma: Does J.S.’ “sane” self have the right to overrule his “insane” self, if the two disagree? Can Dr. Jekyll bind Mr. Hyde? Such a directive is sometimes called a Ulysses contract, after the Greek hero who ordered his men to disregard his commands as they sailed past the sirens. If a patient directs his caregivers to ignore his own future requests, can the caregivers follow these orders?

In J.S.’ case, the answer was no. The research unit’s legal counsel decided that his earlier consent did not override his later refusal, and he was withdrawn from the program. When he resumed his antipsychotic medication and learned what had happened, he begged for another chance to try the experimental medication. Had they been wrong to refuse him?

(Morton E. Winston, Sally M. Winston, Paul S. Appelbaum, and Nancy K. Rhoden, “Can a Subject Consent to a ‘Ulysses Contract’?”, The Hastings Center Report, 12:4 [August 1982], 26-28)

The Six Circles Theorem

Fit a circle into one corner of a triangle. Now fit a second circle into a second corner so that it’s tangent to the first circle. Then fit a third circle into the third corner so that it’s tangent to the second circle.

Keep this up, cycling among the three corners, and the sixth circle will be tangent to the first one.

Dog Tired

Maybe figures can’t lie, but liars can certainly figure, and that is why statistics can be made to ‘prove’ almost anything. Consider a group of ten girls, nine of them virgins, one pregnant. On the ‘average’ each of the nine virgins is ten per cent pregnant, while the girl who is about to have a baby is ninety per cent a virgin. Or, assuming that a fox terrier two feet long, with a tail an inch and a half high, can dig a hole three feet deep in ten minutes, to dig the Panama Canal in a single year would require only one fox terrier fifteen miles long, with a tail a mile and a half high.

— Stuart Cloete, The Third Way, 1947

Never the Twain

proclus paradox

A paradox attributed to Proclus Lycaeus (412-485):

Consider two nonparallel lines, AQ and BP. BP is perpendicular to AB; AQ isn’t. Find the midpoint of AB and mark AC = BD = AB/2. Now if AQ and BP are going to intersect, it can’t happen on AC or BD; if it did, say at a point R, then that would give us a triangle ARB where the sum AR + RB < AB, which is impossible. But now we can connect CD and follow the same process: CE and DF can't intersect for the same reason. EG and FH are likewise ruled out, and so on up the line forever. This seems to mean that two nonparallel lines will never intersect. That can’t be right, but where is the error?

(From Alfred Posamentier, Magnificent Mistakes in Mathematics, 2013.)


Earth is the only planet not named after a god.


Sherlock Holmes is an honorary fellow of the Royal Society of Chemistry.

“Holmes did not exist, but he should have existed,” society chief David Giachardi said in bestowing the award in 2002. “That is how important he is to our culture. We contend that the Sherlock Holmes myth is now so deeply rooted in the national and international psyche through books, films, radio and television that he has almost transcended fictional boundaries.”

The Wheel Cipher

Thomas Jefferson, already absurdly accomplished by 1795, somehow found time to delve into cryptography, where he devised this cipher system. The letters of the alphabet are printed along the rim of each of 36 disks, which are stacked on an axle. One party rotates the disks until his message can be read along one of the 26 rows of letters, somewhat like a modern cylindrical bike lock. Now he can record the letters in any one of the other 25 rows and send that string safely to another party, who decodes it by reversing this procedure. If the message is intercepted, it’s useless even to someone who has the disks, because he must also know the order in which to stack them, and 36 disks can be stacked in 371,993,326,789,901,217,467,999,448,150,835, 200,000,000 different ways.

This is pretty robust. The cipher below, created in 1915 by U.S. Army cryptographer Joseph Mauborgne, has never been solved. “The known systems from this year (or earlier) shouldn’t be too hard to crack with modern attacks and technology,” writes NSA cryptologist Craig P. Bauer. “So, why don’t we have a plaintext yet? My best guess is that it used a cipher wheel” like Jefferson’s.

mauborgne cipher

(L. Kruh, “A 77-year-old challenge cipher,” Cryptologia, 17(2), 172-174, 1993, quoted in Bauer’s Secret History: The Story of Cryptology, 2013.)


Squeeze six circles into a larger circle so that each is tangent to its two neighbors. Now the three lines drawn through opposite points of tangency will pass through the same point.

Remarkably, this wasn’t discovered until 1974.