The Hollowood Function

California high school student Derek Hollowood created this function after considering recurrence relations:

h(x)=\frac{10^{x+1}-9x-10}{81}
h(-10) = 0.987654321
h(-9)  =  0.87654321
h(-8)  =   0.7654321
h(-7)  =    0.654321
h(-6)  =     0.54321
h(-5)  =      0.4321
h(-4)  =       0.321
h(-3)  =        0.21
h(-2)  =         0.1
h(-1)  =           0
h(0)   =           0
h(1)   =           1
h(2)   =           12
h(3)   =           123
h(4)   =           1234
h(5)   =           12345
h(6)   =           123456
h(7)   =           1234567
h(8)   =           12345678
h(9)   =           123456789

(Thanks to Chris Smith for the tip.)

Blind Date

https://pixabay.com/en/clock-wall-clock-watch-time-old-1274699/

When I was 10 years old, a time machine appeared in my bedroom and my older self emerged and tried to kill me. He failed, of course, as his own existence must have shown him he would. But ever since I’ve wondered: This means that someday I myself must travel back to that bedroom and try to kill my younger self. And why would I ever do that?

This is not a logical or a metaphysical problem, but a psychological one. I can imagine being someday so depressed or ashamed or angry at myself that I’m motivated to travel back and try to erase my own existence. But I already know that the gun will jam. What earthly reason, then, could I have to go through the motions of a failed assassination? (Certainly we can imagine cases of amnesia, mistaken identity, etc., where such an action would make sense, but we’re interested in the basic straightforward case in which a time traveler interacts with his younger self — which surely would happen if time travel were possible.)

It seems that I must be motivated, somehow, in order for the appointment to take place, and yet the motivation seems to have no source. “In the present case, we have actions coming from nowhere, in the sense that no one decides, in the usual way, to perform them (or decides that they should be performed), and yet they are performed nonetheless,” writes University of Sydney philosopher Nicholas J.J. Smith. “The psychology of self-interaction is essentially different from that of interaction with others — because the former, but not the latter, involves the problem of agents knowing what they will decide to do, before they decide to do it.”

(Nicholas J.J. Smith, “Why Would Time Travelers Try to Kill Their Younger Selves?”, The Monist 88:3 [July 2005], 388-395.)

Podcast Episode 99: Notes and Queries

https://commons.wikimedia.org/wiki/File:Line5125_-_Flickr_-_NOAA_Photo_Library.jpg

In this week’s episode of the Futility Closet podcast we’ll take a tour through some oddities and unanswered questions from our research, including whether a spider saved Frederick the Great’s life, a statue with the wrong face, and a spectacularly disaster-prone oil tanker.

We’ll also revisit the lost soldiers of World War I and puzzle over some curiously lethal ship cargo.

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Sources for this week’s feature:

The story about Frederick the Great is from Ebenezer Cobham Brewer’s The Reader’s Handbook of Famous Names in Fiction, Allusions, References, Proverbs, Plots, Stories, and Poems, 1899.

The footnote about spiders and flashlights accompanies J.D. Memory’s poem “The Eightfold Way, Lie Algebra, and Spider Hunting in the Dark” in Mathematics Magazine 79:1 (February 2006), 74.

The case of the self-abnegating heir is cited as Beamish v. Beamish, 9 H.L.C. 274, 11 Eng. Rep. 735 (1861) in Peter Suber’s 1990 book The Paradox of Self-Amendment.

John Waterhouse’s 1899 proof of the Pythagorean theorem appears in Elisha Scott Loomis’ 1940 book The Pythagorean Proposition. My notes say it’s also in Scientific American, volume 82, page 356.

The story of the ill-starred oil tanker Argo Merchant is taken from Stephen Pile’s 1979 Book of Heroic Failures. For an exceptionally well-reported history of the ship, see Ron Winslow’s 1978 book Hard Aground.

Physicist Leonard Mlodinow recounts the story of Antoine Lavoisier’s statue in The Upright Thinkers (2015). A contemporary description of the unveiling is here, but it mentions nothing amiss.

Ross Eckler addresses accidental acrostics in Making the Alphabet Dance, 1997.

F.R. Benson’s iambic ponging is mentioned in Jonathan Law, ed., Methuen Drama Dictionary of the Theatre, 2013.

William Kendal’s accomplished blanching is described in Eric Johns’ Dames of the Theatre, 1975.

In The Book of the Harp (2005), John Marson notes that Luigi Ferrari Trecate’s Improvviso da Concerto (1947), for the left hand, is dedicated to harpist Aida Ferretti Orsini, described as grande mutilata di guerre.

Mable LaRose’s 1897 auction is recounted in Pierre Berton’s The Klondike Fever, 2003, and Douglas Fetherling’s The Gold Crusades, 1997.

Listener mail:

Here’s the scene in which the dead of World War I arise in Abel Gance’s 1919 feature J’Accuse:

This week’s lateral thinking puzzle was contributed by listener Price Tipping.

You can listen using the player above, download this episode directly, or subscribe on iTunes or via the RSS feed at http://feedpress.me/futilitycloset.

Many thanks to Doug Ross for the music in this episode.

Enter promo code CLOSET at Harry’s and get $5 off your first order of high-quality razors.

If you have any questions or comments you can reach us at podcast@futilitycloset.com. Thanks for listening!

Magic Space

knecht most-perfect square

Craig Knecht, whose “terraformed” magic squares we explored in 2013, has begun to experiment with applying “magic” properties to David Hilbert’s space-filling curve.

The Hilbert curve finds its way to every cell in the square above by following the pattern shown at the lower left. Knecht divided that path into eight-cell segments, as shown in (a), and then sought solutions in which each colored eight-cell panel produced the magic sum of 260 while each of the eight ordinal positions across the eight panels did so as well. For example, in (b), a large red digit 1 marks the “first” position in each panel; the hope was to find values for these eight cells that would sum to 260, and likewise for all the “second” cells, the “third” ones, and so on.

The result, shown in (c), is a “most-perfect” magic square: Each colored panel sums to 260, and every set of cells that are 8 spaces apart on the Hilbert curve also sum to 260.

The next step was to apply this idea in three dimensions, and recently Knecht made the breakthrough shown below — a 4×4×4 “most perfect” number cube. The 64 numbers in the cube can be broken into 36 2×2 subsquares in each dimension, as shown. In all 108 of these subsquares, the four constituent numbers total 130. And as with the two-dimensional square above, a Hilbert curve can be drawn through the cube that visits each cell once, and cells that are eight cells apart on this curve sum to 260.

One of Knecht’s correspondents pointed out that the cube is even magicker than he had supposed: The “wraparound” subsquares (for example, 5, 28, 44, and 53 on the top of the cube) also sum to 130, as does each set of four corners, making a total of 192 2×2 subsquares that sum to 130.

“So in summary … making the Hilbert space-filling curve path have this magic property of values 8 spaces summing to the magic constant + this 2×2 planar criteria produces a very interesting cube!”

knecht most-perfect cube

Black and White

spencer chess problem

G.B. Spencer devised this ingenious diagram in 1906: It contains 16 separate chess problems, one on each rank and file. In each case, ignore the pieces not on that rank or file and find a way for White to mate in two moves.

For example, the solution to the problem on the first rank is 1. Bd4 Ke2 2. Ng3#. What are the other 15 solutions?

Click for Answer

Viewpoint

https://commons.wikimedia.org/wiki/File:Karl_D%C3%B6nitz.jpg

Captured by the British near the end of World War I, submarine commander Karl Dönitz was a prisoner at Gibraltar when news of the armistice came through. As he and a fellow German officer watched the celebrations “with infinitely bitter hearts,” a British captain joined them on deck.

Dönitz waved his arm in a gesture to encompass all the ships in the roads, British, American, French, Japanese, and asked if he could take any joy from a victory which could only be attained with the whole world for allies.

‘Yes,’ the Captain replied after a pause, ‘it’s very curious.’

In his memoirs, Dönitz wrote, “I will hold the memory of this fair and noble English sea officer in high regard all my life.”

(From Peter Padfield, Donitz: The Last Führer, 2013.)

Parrot Talk

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At his death in 1880, Gustave Flaubert left behind notes for a Dictionary of Received Ideas, a list of the trite utterances that people repeat inevitably in conversation:

  • accident: Always “regrettable” or “unlucky” — as if a mishap might sometimes be a cause for rejoicing.
  • Archimedes: On hearing his name, shout “Eureka!” Or else: “Give me a fulcrum and I will move the world.” There is also Archimedes’ screw, but you are not expected to know what that is.
  • bachelors: All self-centered, all rakes. Should be taxed. Headed for a lonely old age.
  • book: Always too long, regardless of subject.
  • classics: You are supposed to know all about them.
  • forehead: Wide and bald, a sign of genius, or of self-confidence.
  • funny: Should be used on all occasions: “How funny!”
  • gibberish: Foreigners’ way of talking. Always make fun of the foreigner who murders French.
  • handwriting: A neat hand leads to the top. Undecipherable: a sign of deep science, e.g. doctors’ prescriptions.
  • jury: Do everything you can to get off it.
  • metaphors: Always too many in poems. Always too many in anybody’s writing.
  • original: Make fun of everything that is original, hate it, beat it down, annihilate it if you can.
  • pillow: Never use a pillow: it will make you into a hunchback.
  • restaurant: You should order the dishes not usually served at home. When uncertain, look at what others around you are eating.
  • taste: “What is simple is always in good taste.” Always say this to a woman who apologizes for the inadequacy of her dress.
  • war: Thunder against.
  • young gentleman: Always sowing wild oats; he is expected to do so. Astonishment when he doesn’t.

“All our trouble,” he wrote to George Sand, “comes from our gigantic ignorance. … When shall we get over empty speculation and accepted ideas? What should be studied is believed without discussion. Instead of examining, people pontificate.”

(Thanks, Macari.)