In 100 C.E., Nicomachus of Gerasa observed that
13 + 23 + 33 + … + n3 = (1 + 2 + 3 + … + n)2
Or “the sum of the cubes of 1 to n is the same as the square of their sum.” The diagram above demonstrates this neatly: Counting the individual squares shows that
1 × 12 + 2 × 22 + 3 × 32 + 4 × 42 + 5 × 52 + 6 × 62
= 13 + 23 + 33 + 43 + 53 + 63
= (1 + 2 + 3 + 4 + 5 + 6)2
For more than 500 million years something has been making hexagonal burrows on the floor of the deep sea. Each network of tiny holes leads to a system of tunnels under the surface. The creature that makes them, known as Paleodictyon nodosum, has never been discovered. It might be a worm or perhaps a protist; the structure might be its means of farming its own food or the remains of a nest for protecting eggs. Fossils have been found in the limestone of Nevada and Mexico, and the burrows even turn up in the drawings of Leonardo da Vinci. But what makes them, and how, remain a mystery.
Somewhat related: When puzzling screw-shaped structures (below) were unearthed in Nebraska in the 1890s they were known as “devil’s corkscrews” and attributed to freshwater sponges or some sort of coiling plant. They were finally recognized as the burrows of prehistoric beavers only when a fossilized specimen, Palaeocastor, was found inside one.
(Thanks, Paul.)
At Chancellorsville in 1863, a Confederate sniper was plaguing Union troops until an Army sharpshooter thought of a clever solution:
First he took off his cap, and shoved it over the earthwork. Of course, Johnnie Reb let go at it, thinking to kill the careless man under it. His bullet struck into the bank, and instantly our sharpshooter ran his ramrod down the hole made by the Johnnie’s ball, then lay down on his back and sighted along the ramrod. He accordingly perceived from the direction that his game was in the top of a thick bushy elm tree about one hundred yards in the front. It was then the work of less than a second to aim his long telescopic rifle at that tree and crack she went. Down tumbled Mr. Johnnie like a great crow out of his nest, and we had no more trouble from that source.
Recounted in Adrian Gilbert, Sniper, 1994.
We covered one paradox regarding blackmail in 2010: If it’s legal for me to reveal your secret, and it’s legal for me to ask you for money, why is it illegal for me to demand payment to keep your secret? In the words of Northwestern University law professor James Lindgren, “Why do two rights make a wrong?”
Here’s a second paradox: If you had initiated the same transaction — if you had offered to pay me for my silence, and I’d agreed — then we’d have the same outcome, but this time it’s legal. “It is considered paradoxical that the sale of secrecy is legal if it takes the form of a bribe, yet is illegal where the sale of secrecy takes the form of blackmail,” writes Loyola University economist Walter Block. “Why should the legality of a sale of secrecy depend entirely upon who initiates the transaction? Why is bribery legal but blackmail not?”
(Walter Block et al., “The Second Paradox of Blackmail,” Business Ethics Quarterly, July 2000.)
What’s unusual about the number 313,340,350,000,000,000,499? Its English name, THREE HUNDRED THIRTEEN QUINTILLION THREE HUNDRED FORTY QUADRILLION THREE HUNDRED FIFTY TRILLION FOUR HUNDRED NINETY-NINE, contains these letter counts:
This makes the name a perfect “snowball,” in the language of wordplay enthusiasts. In exploring this phenomenon for the November 2012 issue of Word Ways, Eric Harshbarger and Mike Keith found hundreds of thousands of solutions among very large numbers, but the example above is “shockingly small compared to all other known SH [snowball histogram] numbers,” they write. “It seems very likely that this is the smallest SH number of any order, but a proof of this fact, even with computer assistance, seems difficult.”
Two other pretty findings from their article:
224,000,000,000,525,535, or TWO HUNDRED TWENTY-FOUR QUADRILLION FIVE HUNDRED TWENTY-FIVE THOUSAND FIVE HUNDRED THIRTY-FIVE, produces a “growing/melting” snowball:
And 520,636,000,000,757,000, or FIVE HUNDRED TWENTY QUADRILLION SIX HUNDRED THIRTY-SIX TRILLION SEVEN HUNDRED FIFTY-SEVEN THOUSAND, produces the first 18 digits of π:
“This idea can also be applied to arbitrary text, not just number names,” they write. “Can you find a sentence in Moby Dick or Pride and Prejudice whose letter distribution is a snowball or is interesting in some other way? Such possibilities are left for future consideration.”
(Eric Harshbarger and Mike Keith, “Number Names With a Snowball Letter Distribution,” Word Ways, November 2012.)
“Honesty is the best policy: but he who acts on that principle is not an honest man.” — Archbishop Richard Whately
Take two decks of cards, minus the jokers, shuffle them together, and divide them into two piles of 52 cards. What is the probability that the number of red cards in the Pile A equals the number of black cards in Pile B? How many cards would you have to view to be certain of your answer?
The first movement of Beethoven’s piano sonata no. 29, the Hammerklavier, bears a puzzlingly fast tempo marking, half-note=138. Most pianists play it considerably more slowly, judging that the indicated tempo would test the limits of the player’s technique and the listeners’ comprehension.
Well, most listeners. In Fred Hoyle’s 1957 science fiction novel The Black Cloud, an intelligent cloud of gas enters the solar system and establishes communication with the earth. It demonstrates a superhumanly subtle understanding of any information that’s transmitted to it. As scientists are uploading a sampling of Earth music, a lady remarks, “The first movement of the B Flat Sonata bears a metronome marking requiring a quite fantastic pace, far faster than any normal pianist can achieve, certainly faster than I can manage.”
The cloud considers the sonata and says, “Very interesting. Please repeat the first part at a speed increased by thirty percent.”
When this is done, it says, “Better. Very good. I intend to think this over.”
This is not very interesting
But if
You have read this far already
You will
Probably
Read as far as this:
And still
Not really accomplishing
Anything at all
You might
Even read on
Which brings you to
The line you are reading now
And after all that you are still
Probably dumb enough to keep
Right on making
A dope of yourself
By reading
As far down
The page as this.
— Anonymous, Princeton Tiger, 1949
