Expression

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Image: Wikimedia Commons

Eight ways to pronounce the letter X, from wordplay maven Dmitri Borgmann:

eks: x-ray
gz: exist
gzh: luxurious
kris: Xmas
ks: sex
ksh: anxious
z: xylophone
__: faux pas

He adds three more: According to Webster’s Second Edition, xeres is an alternate name for sherry wine in which the X can be pronounced either as H or as SH. And arguably the X in except is pronounced like the letter K, as “the sibilant portion of the usual X sound has fused with the sound of the C immediately following.” If we accept these, then the total rises to 11.

(Dmitri A. Borgmann, “The Ultimate Homonym Group,” Word Ways 17:4 [November 1984], 224-228.)

Surface Matters

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In the 17th century, Italian mathematician Evangelista Torricelli experimented with a figure known as Gabriel’s Horn. Rotate the function y = 1/x about the x-axis for x ≥ 1. The resulting figure has finite volume but infinite surface area — it’s sometimes said that, while the horn could be filled up with π cubic units of paint, an infinite number of square units of paint would be needed to cover its surface.

English cosmologist John D. Barrow describes an infinite wedding cake in which each tier is a solid cylinder 1 unit high; the bottom tier has radius 1, the second radius 1/2, the third radius 1/3, and so on. Now the total volume of the cake is π3/6, but the area of its surface is infinite. Barrow writes, “Our infinite cake recipe requires a finite volume of cake to make but it can never be iced because it has an infinite surface area!”

Mike Steuben, a correspondent of Martin Gardner, imagined a set of boxes, each with area 1 × 1. If the height of the first box is 1, the second 1/2, the third 1/4, and so on, then the total volume of the group is 2 cubic units, but the length and the total area of the tops are infinite.

(Barrow’s example is from 100 Essential Things You Didn’t Know You Didn’t Know About Math and the Arts, 2014.)

Logic and Belief

A syllogism is a logical argument in which a conclusion is inferred from a set of premises:

All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.

The conclusion can be valid without actually stating a true fact; to be valid it just needs to follow logically from the premises. Which of these syllogisms are valid?

No cigarettes are inexpensive.
Some addictive things are inexpensive.
Therefore, some addictive things are not cigarettes.

No addictive things are inexpensive.
Some cigarettes are inexpensive.
Therefore, some cigarettes are not addictive.

In fact both of them are valid. But, interestingly, here the first conclusion seems plausible, while the second does not. That shouldn’t matter, but it does: When Plymouth Polytechnic psychologist J. St. B.T. Evans presented a set of these arguments to subjects in 1983, he found a substantial “belief bias” — the subjects tended to judge the believable conclusions to be valid more than the unbelievable ones. If the conclusion was believable, 92% of the subjects accepted it, regardless of its validity. If the conclusion was unbelievable, 46% accepted it if it was valid, 8% if it was invalid.

Evans wrote, “These findings not only provide a challenge for existing models of syllogistic reasoning but also raise broader questions about people’s rational competence to generate and assess logical arguments in real life, whenever they have clear a priori beliefs about the subject under discussion.”

(J. St. B.T. Evans et al., “On the Conflict Between Logic and Belief in Syllogistic Reasoning,” Memory & Cognition, 11(3), 295-306.)

Tossing and Turning

Butler University mathematician Jerry Farrell has telekinesis. Here’s a demonstration. Toss a coin and enter the result (HEAD or TAIL) as 1 Across in the grid below. Then solve the rest of the puzzle:

farrell puzzle

Across                                   Down

1 Your coin shows a ______               1 Half a laugh
5 Wagner's earth goddess                 2 Station terminus?
6 Word with one or green                 3 Dec follower?
                                         4 Certain male
Click for Answer

Unquote

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“There are very few things which we know, which are not capable of being reduc’d to a Mathematical Reasoning; and when they cannot it’s a sign our knowledge of them is very small and confus’d; and when a Mathematical Reasoning can be had it’s as great a folly to make use of any other, as to grope for a thing in the dark, when you have a Candle standing by you.” — John Arbuthnot, Of the Laws of Chance, 1692

As You Wish

In 1951 G.V. Carey published a 15-page booklet called “Making an Index,” intended to guide new authors in preparing indexes for their books. When it was published, a friendly reviewer in the Times Literary Supplement suggested jokingly that the booklet might have benefited from an index of its own, in which Carey could have given “a full-dress demonstration of his principles.”

So, charmingly, Carey made one: In the second edition he added a 3-page index to his 15-page book, writing, “The reviewer, though he may have had his tongue in his cheek, has put the author on his mettle and tempted him, at the opportunity afforded by a new impression, to take up the challenge.” Admittedly, this required some stretching, particularly as he wanted to include every letter of the alphabet. Some sample entries:

Anybody, mere page-numbers not of the slightest use to, 7
Bibliographer, seventeenth-century, 3
Cherry, twice bitten, once shy. See Cross-references
Common sense, use your, 9, 15, and pass.
Earl of Beaconsfield, 11
Eye in, getting your, 5
Fiction, non-, 3
Haystack, looking for needle in, 4
Jehu (son of Nimshi), 12-13
John, St, 10
Life of Abraham Lincoln, 6
Lincoln, Abraham, Life of, 6
Omniscient, indexers not always, 4
Perfection, counsel of, 3
Sense, common. See Common sense
Suez Crisis, 14
What not to do. See Anybody, Earl of Beaconsfield, von Kluck, etc., etc.
York, New, missing, 10
Yourself in the users’ place, put, 6-7, 12
Zealand, New, 10

One thicket of cross-references never finds its way back to the text:

Chase, wild goose, See Von Kluck
Goose chase, wild. See Kluck, von
Kluck, von. See Von Kluck
Von Kluck. See Kluck, von
Wild goose chase. See Kluck, von

And evidently he hates the word alphabetisation:

Order, alphabetical. See Horrid word
Horrid word. See Alphabetisation
Alphabetisation, 9-10

But “It remains only to affirm that the author has made a serious attempt to demonstrate, even in this not very serious index, some at least of the principles set forth in the preceding pages.”

A New Pangram

British recreational mathematician Lee Sallows has produced many varieties of the self-enumerating pangram — sentences that inventory their own contents:

This pangram contains four As, one B, two Cs, one D, thirty Es, six Fs, five Gs, seven Hs, eleven Is, one J, one K, two Ls, two Ms, eighteen Ns, fifteen Os, two Ps, one Q, five Rs, twenty-seven Ss, eighteen Ts, two Us, seven Vs, eight Ws, two Xs, three Ys, & one Z.

A few years ago he began to wonder whether it’s possible to produce a sentence that reckons its totals as percentages. This is more difficult, because the percentages won’t always work out to be integers. As he worked on the problem he mentioned it to a few others, among them British computer scientist Chris Patuzzo. And a few days ago, Patuzzo sent him this:

This sentence is dedicated to Lee Sallows and to within one decimal place four point five percent of the letters in this sentence are a’s, zero point one percent are b’s, four point three percent are c’s, zero point nine percent are d’s, twenty point one percent are e’s, one point five percent are f’s, zero point four percent are g’s, one point five percent are h’s, six point eight percent are i’s, zero point one percent are j’s, zero point one percent are k’s, one point one percent are l’s, zero point three percent are m’s, twelve point one percent are n’s, eight point one percent are o’s, seven point three percent are p’s, zero point one percent are q’s, nine point nine percent are r’s, five point six percent are s’s, nine point nine percent are t’s, zero point seven percent are u’s, one point four percent are v’s, zero point seven percent are w’s, zero point five percent are x’s, zero point three percent are y’s and one point six percent are z’s.

Details are here. The next challenge is a version where the percentages are accurate to two decimal places — Patuzzo is working on that now.

(Thanks, Lee.)

Podcast Episode 81: The Typhus Hoax

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In 1939, as Germany was sending the people of Poland to labor and death camps, two doctors found a unique way to save their countrymen — by faking an epidemic. In this week’s episode of the Futility Closet podcast we’ll learn about their clever plan, which ultimately saved 8,000 people.

We’ll also consider four schemes involving tiny plots of land and puzzle over why a library would waive its fees for a lost book.

See full show notes …