Richard Feynman tangled regularly with military censors at Los Alamos. Playing one day with a computing machine, he discovered a pleasing little pattern:

1/243 = 0.004115226337448559670781893004115226337448559670781893004115226…

“It’s quite cute, and then it goes a little cockeyed when you’re carrying; confusion occurs for only about three numbers, and then you can see how the 10 10 13 is really equivalent to 114 again, or 115 again, and it keeps on going, and repeats itself nicely after a couple of cycles. I thought it was kind of amusing.”

Well, I put that in the mail, and it comes back to me. It doesn’t go through, and there’s a little note: ‘Look at Paragraph 17B.’ I look at Paragraph 17B. It says, ‘Letters are to be written only in English, Russian, Spanish, Portuguese, Latin, German, and so forth. Permission to use any other language must be obtained in writing.’ And then it said, ‘No codes.’

So I wrote back to the censor a little note included in my letter which said that I feel that of course this cannot be a code, because if you actually do divide 1 by 243 you do, in fact, get all that, and therefore there’s no more information in the number .004115226337… than there is in the number 243 — which is hardly any information at all. And so forth.

“I therefore asked for permission to use Arabic numerals in my letters. So, I got that through all right.”

(From his reminiscences.)


In 1995, Alma College mathematician John F. Putz counted the measures in Mozart’s piano sonatas, comparing the length of the exposition (a) to that of the development and recapitulation (b):

Köchel and movement a b a + b
279, I 38 62 100
279, II 28 46 74
279, III 56 102 158
280, I 56 88 144
280, II 56 88 144
280, II 24 36 60
280, III 77 113 190
281, I 40 69 109
281, II 46 60 106
282, I 15 18 33
282, III 39 63 102
283, I 53 67 120
283, II 14 23 37
283, III 102 171 273
284, I 51 76 127
309, I 58 97 155
311, I 39 73 112
310, I 49 84 133
330, I 58 92 150
330, III 68 103 171
332, I 93 136 229
332, III 90 155 245
333, I 63 102 165
333, II 31 50 81
457, I 74 93 167
533, I 102 137 239
533, II 46 76 122
545, I 28 45 73
547, I 78 118 196
570, I 79 130 209

He found that the ratio of b to a + b tends to match the golden ratio. For example, the first movement of the first sonata is 100 measures long, and of this the development and recapitulation make up 62. “This is a perfect division according to the golden section in the following sense: A 100-measure movement could not be divided any closer (in natural numbers) to the golden section than 38 and 62.”

Ideally there are two ratios that we could hope would hew to the golden section: The first relates the number of measures in the development and recapitulation section to the total number of measures in each movement, and the second relates the length of the exposition to that of the recapitulation and development. The first of these gives a correlation coefficient of 0.99, the second of only 0.938.

So it’s not as impressive as it might be, but it’s still striking. “Perhaps the golden section does, indeed, represent the most pleasing proportion, and perhaps Mozart, through his consummate sense of form, gravitated to it as the perfect balance between extremes,” Putz writes. “It is a romantic thought.”

(John F. Putz, “The Golden Section and the Piano Sonatas of Mozart,” Mathematics Magazine 68:4 [October 1995], 275-282.)


A charming puzzle from Crux Mathematicorum, December 2004:

If all plinks are plonks and some plunks are plinks, which of these statements must be true?

X: All plinks are plunks.
Y: Some plonks are plunks.
Z: Some plinks are not plunks.

Click for Answer


“A little boy and a little girl were looking at a picture of Adam and Eve. ‘Which is Adam and which is Eve?’ said one. ‘I do not know,’ said the other, ‘but I could tell if they had their clothes on.'” — Samuel Butler, Notebooks, 1912

Brave New World

Only a few years back those who carried Umbrellas were held to be legitimate butts. They were old fogies, careful of their health, and so on; but now-a-days we are wiser. Everybody has his Umbrella. It is both cheaper and better made than of old; who, then, so poor he cannot afford one? To see a man going out in the rain umbrella-less excites as much mirth as ever did the sight of those who first — wiser than their generation — availed themselves of this now universal shelter.

— William Sangster, Umbrellas and Their History, 1855

In 1899 Notes & Queries reprinted an account, now thought to be apocryphal, of “the first silk hat in London”:

It was in evidence that Mr. Hetherington, who is well connected, appeared upon the public highway wearing upon his head what he called a silk hat (which was offered in evidence), a tall structure, having a shiny lustre, and calculated to frighten timid people. As a matter of fact, the officers of the Crown stated that several women fainted at the unusual sight, while children screamed, dogs yelped, and a young son of Cordwainer Thomas, who was returning from a chandler’s shop, was thrown down by the crowd which had collected and had his right arm broken.

Supposedly Hetherington argued that he’d broken no law, and the Times backed him up: “In these days of enlightenment it must be considered an advance in dress reform, and one which is bound, sooner or later, to stamp its character upon the entire community.”

The Curse of Billy Penn
Image: Wikimedia Commons

For years, under a “gentleman’s agreement,” the Philadelphia Art Commission would approve no new structure that rose higher than the statue of William Penn atop city hall. Then, in March 1987, it approved One Liberty Place, a steel-and-glass skyscraper that rose 121 meters above Penn’s head.

In the next 22 years no major professional sports team based in Philadelphia won a championship.

Finally, in 2007, during the completion of the 297-meter Comcast Center downtown, workers John Joyce and Dan Ginion attached a small figurine of Penn to its topmost beam. The following year, the Philadelphia Phillies won the World Series.

In 2017 another Penn statuette was placed atop the newly completed 342-meter Comcast Technology Center. “They did not want to take the chance and wait for the jinx,” said the building’s construction manager. A few months later, the Eagles won the Super Bowl.

A Glass Darkly

In “The Adventure of the Empty House,” an old book collector visits John Watson’s house, “his precious volumes, a dozen of them at least, wedged under his right arm.” He says, “Maybe you collect yourself, sir; here’s British Birds, and Catullus, and The Holy War — a bargain every one of them. With five volumes you could just fill that gap on that second shelf.”

Now, this must mean either that two of the titles comprised two volumes apiece or that one comprised three volumes, a point first made by Magistrate S. Tupper Bigelow. But which is it? In the strangely half-specified world that Holmes and Watson inhabit, the fact of the matter seems not to exist.

Philosopher Terence Parsons asks whether Holmes has a mole on his back. Since the stories are silent on this point, it seems that he neither has one nor doesn’t.

See Truth and Fiction.