Buttoned Up

Divide a pile of 14 buttons into two smaller piles, say of 9 and 5 buttons. Then write on a piece of paper: 9 × 5 = 45. Divide the pile of 9 into two smaller piles, say of 6 and 3, and write 6 × 3 = 18 on the paper. Keeping doing this, splitting each pile into two and recording the pair of numbers you get, until you have 14 separate piles of one button each. An example might run like this:

9 × 5 = 45

6 × 3 = 18
1 × 4 = 4

4 × 2 = 8
2 × 1 = 2
2 × 2 = 4

3 × 1 = 3
1 × 1 = 1
1 × 1 = 1
1 × 1 = 1
1 × 1 = 1

1 × 2 = 2

1 × 1 = 1

No matter how you proceed, if you start with a pile of 14 buttons, the products in the right column will always sum to 91.

(James Tanton, “A Dozen Questions About Pile Splitting,” Math Horizons 12:1 [September 2004], 28-31.)

Procrustes

Rhymes for unrhymable words, by Willard R. Espy:

It is unth-
inkable to find
A rhyme for month
Except this special kind.

The four eng-
ineers
Wore orange
Brassieres.

Love’s lost its glow?
No need to lie; j-
ust tell me “go!”
And I’ll oblige.

(From his entertaining rhyming dictionary.)

Flashes

Striking excerpts from the writings of Scottish novelist Muriel Spark, from Penelope Jardine’s 2018 collection A Good Comb:

  • The superstition of today is the science of yesterday.
  • Providers are often disliked, often despised.
  • I think “waiter” is such a funny word. It is we who wait.
  • It is impossible to persuade a man who does not disagree, but smiles.
  • I’m not lonely before they come. I’m only lonely when they go away.
  • Dangerous people often seem boring.
  • She was astonishingly ugly, one was compelled to look at her.
  • I am an honest man … when treating of the few existing subjects to which honesty is due.
  • Suffering isn’t in proportion to what the sufferer deserves.
  • He exhausted his capacity for conversation when he became an Englishman.

Jardine’s title comes from the observation “It calms you down, a good comb,” remarked by an unnamed character in Spark’s 1960 novel The Ballad of Peckham Rye.

“Coal Is Decayed Vegetarians”

Memorable excerpts from student geology examinations, from W.D. Ian Rolfe’s 1980 collection Geological Howlers:

  • The average person does not have to dig a deep hole to remind himself of the past.
  • Dust is mud with the juice squeezed out.
  • Articulate brachiopods have teeth and socks.
  • A skeleton is a man with his inside out and his outside off.
  • There are three kinds of rocks, ingenious, sedentary and metaphoric.
  • The term Caledonian Orogeny is brandished about by many people.
  • Nine-eighths of an iceberg is beneath the sea.
  • It has been found by a gentle man that organic remains can be converted to petroleum by the processes of metabolism.
  • Sedimentation is a rather lengthy affair.

“A dinosaur is an extinct animal still found in Australia,” one student contended. “It was sometimes so large that its feet are found in the Precambrian and its head in the Silurian because it was too big to lie down where it died.”

Never Mind

After making an appeal for recruits in 1914, Secretary of State for War Lord Kitchener received this letter from one Alfie Knight of Dublin:

Dear Lord Kitchner,

I am an Irish boy 9 years of age and I want to go to the front. I can ride jolley quick on my bicycle and would go as despatch ridder. I wouldint let the germans get it. I am a good shot with a revolver and would kill a good vue of the germans. I am very strong and often win a fight with lads twice as big as mysels. I want a uneform and a revolver and will give a good acount of myself. Pleese send an anencer.

Yours affectionately

Alfie Knight

The War Office sent him this reply:

Lord Kitchener asks me to thank you for your letter, but he is afraid that you are not yet quite old enough to go to the front as a dispatch rider.

(From the Imperial War Museum.)

No Time Like …

In Book 11 of the Confessions, Augustine writes, “Are the present hundred years a long time? But first see whether a hundred years can be present. If it is the first year of the hundred, then that year is present, but the other ninety-nine are still in the future, and so as yet are not; if we are in the second year, then one year is past, one year is present, the rest future. Thus whichever year of our hundred-year period we choose as present, those before it have passed away, those after it are still to come. Thus a hundred years cannot be present.”

Is the chosen year itself present? Not wholly: We’re in some particular month, and the other months are not present. And so on — Augustine applies the same argument to days, hours, and even “fleeting moments.” In the end, “If we conceive of some point of time which cannot be divided even into the minutest parts of moments, that is the only point that can be called present: and that point flees at such lightning speed from being future to being past, that it has no extent of duration at all. For if it were so extended, it would be divisible into past and future: the present has no length.”

Unable Was I

https://commons.wikimedia.org/wiki/File:Opere_varie_Manzoni_1881-695.1.png

It sounds straightforward to imagine being another person, but is it? If I want to imagine being Napoleon, I need to conceive some relation between our two identities. If I only imagine some situation that was faced by Napoleon, then the result involves too little of my own identity — I’m not really involved at all. But if I imagine myself in Napoleon’s place, then the result involves too little of him. It doesn’t seem possible for two people to share an identity in this way.

Philosopher Bernard Williams writes, “Leibniz, perhaps, made something like this point when he said to one who expressed the wish that he were King of China, that all he wanted was that he should cease to exist and there should be a King in China.”

But, Williams says, it does seem possible to play a role, to pretend to be Napoleon. In that case my first-person thoughts are framed in another’s point of view, so the identity of “I” is less problematic. In this sense perhaps I can imagine being Napoleon — but not having been Napoleon.

(Bernard Williams, Problems of the Self, 1976.)

Getting Started

In 1988, tunneling operations began in both England and France. On Dec. 1, 1990, these two single-entrance holes met under the English Channel, after which there existed a two-entrance tunnel. The completed Channel Tunnel would not be ready for use until 1994. Now, suppose that a speaker in 1989 had said:

There is in the process of coming into existence, so we understand, the Channel Tunnel. Not many of us have seen it; I certainly have not. One understands that that will allow us, if and when it is made available, to travel by train from England to France.

Does this utterance imply that, at the time it was spoken, the Channel Tunnel already existed? If not, what is the it that the speaker says few people have seen? In 1989 all that existed were two blind tunnels, which together could not permit free passage between England and France, surely an essential feature of the Channel Tunnel.

“In the construction of the Channel Tunnel, the time at which we can say that a two-entrance tunnel first exists is 1st December 1990, when the two one-entrance tunnels met,” notes philosopher Antony Galton. “The Channel Tunnel is a two-entrance tunnel; so is this time, 1st December 1990, also the time at which the Channel Tunnel first exists?”

(Antony Galton, “On the Process of Coming Into Existence,” Monist 89:3 [July 2006], 294-312.)