The Ulam Spiral

Write the numbers from 41 to 440 in a square spiral:

ulam spiral

Remarkably, all the numbers on the red diagonal are prime — even when the spiral is continued into a 20 × 20 square.

No one’s quite sure what to make of this. Polish mathematician Stanislaw Ulam discovered the pattern while doodling at a scientific meeting in 1963.

A Bad Night

Description of the bed chamber of countess Cornelia Bandi as discovered by her maid one morning in 1731, reprinted in the Philosophical Transactions of the Royal Society of London, 1745:

Four feet distance from the bed there was a heap of ashes, 2 legs untouched, from the foot to the knee, with their stockings on: between them was the lady’s head: whose brains, half of the back part of the skull, and the whole chin, were burnt to ashes; among which were found 3 fingers blackened. All the rest was ashes, which had this particular quality, that they left in the hand, when taken up, a greasy and stinking moisture.

… The bed received no damage; the blankets and sheets were only raised on one side, as when a person rises up from it, or goes in; the whole furniture, as well as the bed, was spread over with moist and ash-coloured soot, which had penetrated into the chest-of-drawers, even to foul the linens; nay the soot was also gone into a neighbouring kitchen, and hung on the walls, moveables, and utensils of it. From the pantry a piece of bread covered with that soot, and grown black, was given to several dogs, which refused to eat it.

“It is impossible that by any accident the lamp should have caused such a conflagration,” remarks the correspondent. “There is no room to suppose any supernatural cause. The likeliest cause then is a flash of lightning.”

Sighs and Whispers

http://www.google.com/patents?id=PM9bAAAAEBAJ&printsec=abstract&zoom=4#v=onepage&q=&f=false

Frustrated with the intertitles in silent films, Charles Pidgin invented a better solution in 1917: The performers would inflate balloons on which their dialogue was printed. “The blowing or inflation of the devices by the various characters of a photo-play will add to the realism of the picture by the words appearing to come from the mouth of the players,” Pidgin wrote. Even better, “the size of the speech may be increased with the increase of various emotions depicted on the screen.”

It’s not too late to implement this.

Cookie-Cutter Holes

On Nov. 24, 1984, the Spokane, Wash., Spokesman-Review reported the discovery of a massive chunk of earth, 10 feet long by 7 feet wide, that had somehow been plucked from the ground and put down, right side up and intact, 73 feet away. Roots had been torn apart rather than cut, and, strangely, the debris between the hole and the slab traced an arc rather than a straight line.

“All we know for sure is that this puzzle piece of earth is 73 feet away from the hole it came out of,” said geologist Greg Behrens.

Similar “cookie-cutter holes” have been observed elsewhere; the earliest known reference is in the Royal Frankish Annals of the 8th century:

In the land of the Thuringians, in the neighborhood of a river, a block of earth fifty feet long, fourteen feet wide, and a foot and a half thick, was cut out, mysteriously lifted, and shifted twenty-five feet from its original location.

No doubt there’s a mundane explanation for this, but for now no one knows what it is.

One-Note Sentences

Monotonous conversation from around the world:

Finnish:

Etsivät etsivät etsivät etsivät etsivät.
“The searching detectives are searching the searching detectives.”

Spanish:

¿Cómo como? ¿Cómo cómo como? Como como como.
“How do I eat? What do you mean, how do I eat? I eat how I eat.”

Icelandic:

Ái á Á á á í á.
“A farmer named Ái, who lives on a farm by the name of Á, owns a female sheep that is in a river.”

Malay:

Sayang, sayang sayang sayang, sayang sayang sayang?
“Darling, I love you, dear, do you love me?”

Romanian:

Stanca sta-n castan ca Stan.
“Stanca stood in a chestnut tree like Stan.”

Hungarian:

A követ követ követ.
“The envoy follows a stone.”

Tagalog:

Bababa ba? Bababa!
“Going down? It is!”

The Self-Flowing Flask

http://commons.wikimedia.org/wiki/File:Boyle'sSelfFlowingFlask.png

Denis Papin described this perpetual motion scheme in 1685. The pint of liquid in the goblet weighs more than the ounce in the tube, and so forces it down. When the liquid spills out the top of the tube and back into the goblet, it starts a cycle that must continue until the liquid finally evaporates. Where is the error?

Pieces of Pi

In the decimal expansion of π:

  • the digits 27182818 — the first eight digits of e — appear at position 1,526,800.
  • the digits 14142135 — the first eight digits of the square root of 2 — appear at position 52,638.
  • the first eight digits of π itself — 31415926 — reappear at position 50,366,472.
  • 16470 appears at position 16470.
  • there are seven 7s at position 3,346,228, eight 8s at position 46,663,520, and six 9s at position 762.
  • The White House switchboard number (456-1414) is at position 3,193,808, the population of France (65,073,482) is at position 98,709,092, and Disneyland’s zip code (92802) is at position 41,112.

Write out the alphabet starting with J:

JKLMNOPQRSTUVWXYZABCDEFGHI

Erase all letters that have left-right symmetry (such as A) and count the letters in each of the five groups that remain. (James Davis)