Langford’s Problem

https://commons.wikimedia.org/wiki/File:Langford_pairing.svg

In the 1950s, mathematician C. Dudley Langford was watching his son play with blocks, two of each color, when he noticed that they formed a curious pattern: one block lay between the red blocks, two between the blue blocks, and three between the yellow blocks. Langford found that by rearranging the blocks he could add a green pair with four blocks between them.

This presented a clear challenge. He found solutions for as many as 15 pairs of blocks but came to believe that some smaller groupings (14 pairs, for example) could not produce a solution. He asked for a general investigation.

Today we know that a solution exists if and only if the number of blocks is 4k or 4k + 3, so Langford was right that no solution can be arranged with 14 pairs of blocks. The number of solutions for each quantity of pairs is listed here, and a few proofs are given here.

(C. Dudley Langford, “Problem,” Mathematical Gazette 42:341 [October 1958], 228.)

| Science & Math

Still Waiting

In a 1901 parody edition, the journal Mind! offered £1,000 to any philosopher who could produce adequate documentary evidence that he:

  1. Knows what he means.
  2. Knows what anyone else means.
  3. Knows what everyone means.
  4. Knows what anything means.
  5. Knows what everything else means.
  6. Means what he says.
  7. Means what he means.
  8. Means what everyone else means.
  9. Means what everyone else says that he means.
  10. Can express what he means.
  11. Knows what it signifies what he means.
  12. Knows what it matters what he signifies.

“At first sight it might seem as though the Twelve Labours of Hercules would be in comparison with this a slighter achievement,” the editors wrote. “But in view of the extensive and peculiar knowledge of the Absolute’s Mind which is now possessed by so many philosophers, a large number of solutions may confidently be expected.”

| Humor

Oh, Never Mind

A man of the State of Chêng was one day gathering fuel, when he came across a startled deer, which he pursued and killed. Fearing lest any one should see him, he hastily concealed the carcass in a ditch and covered it with plantain leaves, rejoicing excessively at his good fortune. By and by, he forgot the place where he had put it, and, thinking he must have been dreaming, he set off towards home, humming over the affair on his way.

Meanwhile, a man who had overheard his words, acted upon them, and went and got the deer. The latter, when he reached his house, told his wife, saying, ‘A woodman dreamt he had got a deer, but he did not know where it was. Now I have got the deer; so his dream was a reality.’ ‘It is you,’ replied his wife, ‘who have been dreaming you saw a woodman. Did he get the deer? and is there really such a person? It is you who have got the deer: how, then, can his dream be a reality?’ ‘It is true,’ assented the husband, ‘that I have got the deer. It is therefore of little importance whether the woodman dreamt the deer or I dreamt the woodman.’

Now when the woodman reached his home, he became much annoyed at the loss of the deer; and in the night he actually dreamt where the deer then was, and who had got it. So next morning he proceeded to the place indicated in his dream, — and there it was. He then took legal steps to recover possession; and when the case came on, the magistrate delivered the following judgment:– ‘The plaintiff began with a real deer and an alleged dream. He now comes forward with a real dream and an alleged deer. The defendant really got the deer which plaintiff said he dreamt, and is now trying to keep it; while, according to his wife, both the woodman and the deer are but the figments of a dream, so that no one got the deer at all. However, here is a deer, which you had better divide between you.’

— Herbert Allen Giles, A History of Chinese Literature, 1927

| Oddities

A Code Poem

On April 1, 1990, an anonymous verse was posted to the comp.lang.perl newsgroup on Usenet. It was written in the programming language Perl 3:

BEFOREHAND: close door, each window & exit; wait until time.
    open spellbook, study, read (scan, select, tell us);
write it, print the hex while each watches,
    reverse its length, write again;
    kill spiders, pop them, chop, split, kill them.
        unlink arms, shift, wait & listen (listening, wait),
sort the flock (then, warn the "goats" & kill the "sheep");
    kill them, dump qualms, shift moralities,
    values aside, each one;
        die sheep! die to reverse the system
        you accept (reject, respect);
next step,
    kill the next sacrifice, each sacrifice,
    wait, redo ritual until "all the spirits are pleased";
    do it ("as they say").
do it(*everyone***must***participate***in***forbidden**s*e*x*).
return last victim; package body;
    exit crypt (time, times & "half a time") & close it,
    select (quickly) & warn your next victim;
AFTERWORDS: tell nobody.
    wait, wait until time;
    wait until next year, next decade;
        sleep, sleep, die yourself,
        die at last

Because of the large number of English words that are used in the Perl language, the poem can actually be compiled as legal code and executed as a program. (It exits on line one, reaching the function exit, producing no output.)

The poem was attributed to “a person who wishes to remain anonymous,” but new “Perl poems” are regularly submitted to the programming community at PerlMonks.

| Poems · Technology

Nothing Ventured

A dentist spotted a deadbeat patient while dining at his country club one evening. He called the patient aside, reminded him that he owed him $250 for work done more than two years earlier, and insisted the man pay up. To the dentist’s astonishment, the patient pulled a checkbook from his pocket and wrote a check to the dentist for the full amount.

Skeptical about the patient’s good faith, the dentist went directly to the bank the next morning and presented the check for payment. The teller handed it back with the explanation that the patient’s account was a little short of the amount of the check. Following a few minutes of good-natured conversation the dentist learned that the man’s account was twenty-five dollars short of the needed amount. The dentist smiled, went to the customers’ desk for a few minutes, came back to the teller, deposited thirty dollars to the account of the patient, and then again presented the check for $250 and walked out with a net gain of $220.

— Ralph L. Woods, How to Torture Your Mind, 1969

| Oddities

In a Word

philomath
n. a lover of learning; a scholar

noddary
n. a foolish act

fedifragous
adj. contract-breaking

subitaneous
adj. sudden

In Fredric Brown’s 1954 short short story “Experiment,” Professor Johnson displays a brass cube and proposes to send it backward in time. An identical cube appears in his time machine at 2:55, and Johnson announces that at 3:00 he will complete the transaction by sending his own cube 5 minutes into the past.

A friend asks: What would happen if he changed his mind and didn’t send it?

“An interesting idea,” says the professor. “I had not thought of it, and it will be interesting to try. Very well, I shall not …”

“There was no paradox at all. The cube remained.

“But the entire rest of the Universe, professors and all, vanished.”

| Language · Literature

Query

Jones had been greatly depressed; he declared himself a murderer, and would not be comforted. Suddenly he asked me a question. ‘Are not the parents the cause of the birth of their children?’ said he. ‘I suppose so,’ said I. ‘Are not all men mortal?’ ‘That also may be admitted.’ ‘Then are not the parents the cause of the death of their children, since they know that they are mortal? And am I not a murderer?’ I was, I own, puzzled. At last I thought of something soothing. I pointed out to Jones that to cause the death of another was not necessarily murder. It might be manslaughter or justifiable homicide. ‘Of which of these then am I guilty?’ he queried. I could not say because I had never seen the Jones family, but I hear Jones has become a great bore in the asylum by his unceasing appeals to every one to tell him whether he has committed murder, manslaughter, or justifiable homicide!

— Rueben Abel, ed., Humanistic Pragmatism: The Philosophy of F.C.S. Schiller, 1966

| Death · Oddities · Society