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

Gesundheit

https://commons.wikimedia.org/wiki/File:Nasothek-2.jpg
Image: Wikimedia Commons

Classical statues tend to lose their noses, and in the 19th century museums would commonly replace them with “restoration” noses, to preserve the appearance of the original sculpture.

In the 20th century some museums changed philosophies and “de-restored” their collections, thinking it better to present each piece in its authentic state.

This created a superfluity of noses, and some museums collect these into displays of their own. Charmingly, there’s even a word for this: A collection of noses is a Nasothek.

Above is the collection in the Ny Carlsberg Glyptotek museum in Copenhagen.

(Thanks, Carsten.)

AWOL

https://commons.wikimedia.org/wiki/File:Christ_Church,_Oxford_2.JPG

A harmonic progression is a progression formed by taking the reciprocals of an arithmetic progression (so an example is 1/1, 1/2, 1/3, 1/4 …). When tutoring mathematics at Oxford, Charles Dodgson had a favorite example to illustrate this:

According to him, it is (or was) the rule at Christ Church that, if an undergraduate is absent for a night during term-time without leave, he is for the first offence sent down for a term; if he commits the offence a second time, he is sent down for two terms; if a third time, Christ Church knows him no more. This last calamity Dodgson designated as ‘infinite.’ Here, then, the three degrees of punishment may be reckoned as 1, 2, infinity. These three figures represent three terms in an ascending series of Harmonic Progression, being the counterparts of 1, 1/2, 0, which are three terms in a descending Arithmetical Progression.

— Lionel A. Tollemache, “Reminiscences of ‘Lewis Carroll,'” Literature, Feb. 5, 1898

Books

FC book covers

Just a reminder — Futility Closet books make great gifts for people who are impossible to buy gifts for. Both contain hundreds of hand-picked favorites from our 12-year archive of curiosities. Some reviews:

“A wild, wonderful, and educational romp through history, science, zany patents, math puzzles, wonderful words (like boanthropy, hallelujatic, and andabatarian), the Devil’s Game, self-contradicting words, and so much more. Buy this book and feed your mind!” — Clifford A. Pickover, author of The Mathematics Devotional

“Futility Closet delivers concentrated doses of weird, wonderful, brain-stimulating ideas and anecdotes, curated mainly from forgotten old books. I’m hooked — there’s nothing quite like it!” — Mark Frauenfelder, founder, Boing Boing

“Meant to be read in pieces, but impossible to put down.” — Gary Antonick, editor, New York Times Numberplay blog

“Futility Closet is a dusty museum back room where one can spend minutes or hours among seldom-seen curiosities, and feel that none of the time was wasted.” — Alan Bellows, DamnInteresting.com

Both books are available now on Amazon. Thanks for your support!

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Reminder

This site is supported entirely by its readers. If you value Futility Closet, please consider making a contribution to keep us going.

You can make a one-time donation on our Support Us page.

And you can get bonus posts and other rewards by pledging a monthly donation on Patreon — you choose the amount to contribute, and you can change or cancel it at any time.

Thanks to everyone who has contributed!

Reminder

This site is supported entirely by its readers. If you value Futility Closet, please consider making a contribution to keep us going.

You can make a one-time donation on our Support Us page.

And you can get bonus posts and other rewards by pledging a monthly donation on Patreon — you choose the amount to contribute, and you can change or cancel it at any time.

Thanks to everyone who has contributed!