Picket Fences

https://commons.wikimedia.org/wiki/File:First_six_triangular_numbers.svg — Image: Wikimedia Commons

A triangular number is one that counts the number of objects in an equilateral triangle, as above:

1
1 + 2 = 3
1 + 2 + 3 = 6
1 + 2 + 3 + 4 = 10
1 + 2 + 3 + 4 + 5 = 15
1 + 2 + 3 + 4 + 5 + 6 = 21

Some of these numbers are palindromes, numbers that read the same backward and forward. A few examples are 55, 66, 171, 595, and 666. In 1973, Charles Trigg found that of the triangular numbers less than 151340, 27 are palindromes.

But interestingly, every string of 1s:

1
11
111
1111
11111
…

… is a palindromic triangular number in base nine. For example:

11₉ = 9 + 1 = 10
111₉ = 9² + 9 + 1 = 91
1111₉ = 9³ + 9² + 9 + 1 = 820
11111₉ = 9⁴ + 9³ + 9² + 9 + 1 = 7381

The pattern continues — all these numbers are triangular.

02/12/2017 UPDATE: Reader Jacob Bandes-Storch sent a visual proof:

“Given a number n in base 9, if we tack a 1 on the right, the resulting number is 9*n + 1. (By shifting over one place to the left, each digit becomes nine times its original value, and then we add 1 in the ones place.) So given a triangular number, there’s probably a way of sticking together 9 copies of it with a single additional unit to form a new triangle. Sure enough:”

bandes-storch proof 1

bandes-storch proof 2

February 11, 2017February 22, 2017 | Science & Math

R.I.P. Raymond Smullyan, 1919–2017

Philosopher and logician Raymond Smullyan passed away on Monday. He was 97.

From my notes, here’s a paradox he offered at a Copenhagen self-reference conference in 2002:

Have you heard of the LAA computing company? Do you know what LAA stands for? It stands for ‘lacking an acronym.’

Actually, the above acronym is not paradoxical; it is simply false. I thought of the following variant which is paradoxical — it is the LACA company. Here LACA stands for ‘lacking a correct acronym.’ Assuming that the company has no other acronym, that acronym is easily seen to be true if and only if it is false.

February 10, 2017 | Science & Math

Small World

paramount shooting locations 1927

In 1927, Chicago investment bank Halsey, Stuart & Co. published a prospectus inviting its clients to consider investing in the burgeoning motion picture business. Among the illustrations was this Paramount Studios map of international shooting locations in California.

“It was not mere chance that established the motion picture industry in Southern California,” notes the booklet. “The actual localization of production there came through a process of pure competitive selection in which the geographical advantages favorable to producers in that region literally forced competing directors and their companies to come to California — and Hollywood.

“The advantages of dependable sunshine, permitting outdoor production without delays, and of great variety in scenery at close range (the ocean on one side, the deserts, mountains, and forests in other directions), so that the sequence of almost any picture can be suitably filmed with but little cost for travel — all have militated to established Hollywood as the center of the motion picture world.”

(“The Motion Picture Industry as a Basis for Bond Financing,” reprinted in Tino Balio, ed., The American Film Industry, 1985.)

February 10, 2017February 9, 2017 | Entertainment

The Windmill Algorithm

Suppose we have a finite set of points in the plane, no three of which are collinear. A line drawn through one of them pivots around that point until it encounters another point, when it adopts that point as the new pivot. Call this line a “windmill”; it continues indefinitely, always rotating in the same direction. Show that we can choose an initial point and line so that the resulting windmill uses each point as a pivot infinitely many times.

	Select Click for Answer
Draw the line vertically through the centermost of the points, and color all the points to the left of it blue and all the points to the right red. After the line has turned 180 degrees, all the points to the left of it will now be red, and those to the right will be blue. Hence every point has changed color. Each time the pivot changes, the old pivot point returns to the same side of the line that the new one comes from. That means that the total number of blue points remains unchanged; so does the number of red points. Since a point can change color only by serving as a pivot, this means that every point must act as a pivot at some stage. And if the line turns forever, then each point will change color infinitely many times. This is laid out more rigorously here. By Geoffrey Smith of the United Kingdom for the 52st International Mathematical Olympiad in Amsterdam, 2011. The animation is from the excellent Symmetry tumblr.

Click for Answer

February 9, 2017February 19, 2017 | Puzzles

Art Appreciation

Vernet relates, that he was once employed to paint a landscape, with a cave, and St. Jerome in it; he accordingly painted the landscape, with St. Jerome at the entrance of the cave. When he delivered the picture, the purchaser, who understood nothing of perspective, said, ‘the landscape and the cave are well made, but St. Jerome is not in the cave.’ ‘I understand you, Sir,’ replied Vernet, ‘I will alter it.’ He therefore took the painting and made the shade darker, so that the saint seemed to sit farther in. The gentleman took the painting; but it again appeared to him that the saint was not in the cave. Vernet then wiped out the figure, and gave it to the gentleman, who seemed perfectly satisfied. Whenever he saw strangers to whom he shewed the picture, he said, ‘Here you see a picture by Vernet, with St. Jerome in the cave.’ ‘But we cannot see the saint,’ replied the visitors. ‘Excuse me, gentlemen,’ answered the possessor, ‘he is there; for I have seen him standing at the entrance, and afterwards farther back; and am therefore quite sure that he is in it.’

— Thomas Byerley and Joseph Clinton Robertson, The Percy Anecdotes, 1821

February 8, 2017February 8, 2017 | Art

Split Decision

In December 2013 a U.S. District Court decided that copyright in the fictional characters Sherlock Holmes and Dr. Watson had expired, but only for the characters as they’re depicted in the earlier novels by Arthur Conan Doyle. Aspects of the characters that are mentioned only in the later novels — such as Dr. Watson’s athletic background, first described in a 1924 short story — are considered new “increments of expression” of those characters, and remain protected.

That makes eminent sense for writers and lawyers, but what about poor Dr. Watson, anxiously stirring the fire at 221B Baker Street? Does he have an athletic background or doesn’t he? The copyright law seems to apply to a version of him that does, and not to one that doesn’t. Should we say there are two Dr. Watsons? That doesn’t seem right.

Worse, “If an author now wants to write a new Holmes novel, but is prohibited from mentioning almost everything pertaining to Professor Moriarty (who only rose to prominence in the later work Valley of Fear), how can we say that he is still writing about the ‘the same’ Holmes, given how much his character was formed through the interaction with his nemesis?” ask legal scholars Burkhard Schafer and Jane Cornwell. “Does this not render any new Holmes necessarily ‘incomplete,’ that is lacking character traits and memories Holmes is ‘known to’ possess, according to the canonical work?”

Even the “public domain” Holmes seems to multiply in this light. We learn that Holmes has an older brother, Mycroft, in “The Adventure of the Greek Interpreter,” published in 1893. But if Mycroft is older than Sherlock, then surely he’s been Sherlock’s brother ever since Sherlock’s birth in 1854. Does the early Sherlock (in, say, A Study in Scarlet) have a brother?

(Burkhard Schafer and Jane Cornwell, “Law’s Fictions, Legal Fictions and Copyright Law,” in Maksymilian Del Mar and William Twining, eds., Legal Fictions in Theory and Practice, 2015.)

February 8, 2017February 7, 2017 | Literature · Oddities

Noted

In Pascal’s triangle, above, the number in each cell is the sum of the two immediately above it.

If you “tilt” the triangle so that each row starts one column to the right of its predecessor, then the column totals produce the Fibonacci sequence:

pascal triangle fibonacci numbers

That’s from Thomas Koshy’s Triangular Arrays With Applications, 2011.

Bonus: Displace the rows still further and they’ll identify prime numbers.

February 7, 2017 | Science & Math

Desperate Measures

From reader Jon Sweitzer-Lamme:

Pressed for materials during the siege of Vicksburg in 1863, the city’s Daily Citizen newspaper printed its last six issues on the back of wallpaper. Each of the issues for June 16, 18, 20, 27, 30, and July 2 was printed in four columns on a single sheet, as above; a reader who turned the sheet over would see this:

The last issue, on July 2, is still defiant:

The Yankee Generalissimo surnamed Grant has expressed his intention of dining in Vicksburg on the Fourth of July. … Ulysses must get into the city before he dines in it.

But two days later, when the city finally fell, Union troops added a final paragraph:

Two days bring about great changes, The banner of the Union floats over Vicksburg. Gen. Grant has ‘caught the rabbit:’ he has dined in Vicksburg, and he did bring his dinner with him. The ‘Citizen’ lives to see it. For the last time it appears on ‘Wall-paper.’ No more will it eulogize the luxury of mule-meat and fricassed kitten — urge Southern warriors to such diet never-more. This is the last wall-paper edition, and is, excepting this note, from the types as we found them. It will be valuable hereafter as a curiosity.

More at the Library of Congress. (Thanks, Jon.)

February 7, 2017February 6, 2017 | History · Oddities

Unquote

“All the glory of the world would be buried in oblivion, unless God had provided mortals the remedy of books.” — Richard de Bury

February 6, 2017 | Quotations

Podcast Episode 140: Ramanujan

In 1913, English mathematician G.H. Hardy received a package from an unknown accounting clerk in India, with nine pages of mathematical results that he found “scarcely possible to believe.” In this week’s episode of the Futility Closet podcast, we’ll follow the unlikely friendship that sprang up between Hardy and Srinivasa Ramanujan, whom Hardy called “the most romantic figure in the recent history of mathematics.”

We’ll also probe Carson McCullers’ heart and puzzle over a well-proportioned amputee.

See full show notes …

February 6, 2017October 1, 2017 | Podcast · Science & Math