The so-called four-field approach in anthropology divides the discipline into four subfields: archaeology, linguistics, physical anthropology, and cultural anthropology.
Students call these “stones, tones, bones, and thrones.”
The so-called four-field approach in anthropology divides the discipline into four subfields: archaeology, linguistics, physical anthropology, and cultural anthropology.
Students call these “stones, tones, bones, and thrones.”
A centered hexagonal number is a number that can be represented by a hexagonal lattice with a dot in the center, like so:
Starting at the center, successive hexagons contain 1, 7, 19, and 37 dots. The sequence goes on forever.
The sum of the first n centered hexagonal numbers is n3, and there’s a pretty “proof without words” to show that this is so:
Instead of regarding each figure as a hexagon, think of it as a perspective view of a cube, looking down along a space diagonal. The first cube here contains a single dot. How many dots must we add to produce the next larger cube? Seven, and from our bird’s-eye perspective this pattern of 7 added dots matches the 7-dot hexagon shown above. The same thing happens when we advance to a 3×3×3 cube: This requires surrounding the 2×2×2 cube with 19 additional dots, and from our imagined vantage point these again take the form of a hexagonal lattice. In the last image our 33 cube must accrete another 37 dots to become a 43 cube … and the pattern continues.
Reader Derek Christie sent in this surprising curiosity after Wednesday’s post about Borromean rings:
Both the ring and the karabiner clip are attached to the cord and can’t be removed.
Now complicate matters by clipping the karabiner onto the ring:
Quite unexpectedly, the cord can now be just pulled away:
(Thanks, Derek.) (A related perplexity: The Prisoners’ Release Puzzle.)
“Rules for the direction of the mind,” from an unfinished treatise by René Descartes:
He’d planned a further 15 but did not finish the work. These 21 were published posthumously in 1701.
It is time to bury the nonsense of the ‘incomplete animal.’ As Julian Huxley, the eminent British biologist, once observed concerning human toughness, man is the only creature that can walk twenty miles, run two miles, swim a river, and then climb a tree. Physiologically, he has one of the toughest bodies known; no other species could survive weeks of exposure on the open sea, or in deserts, or the Arctic. Man’s superior exploits are not evidence of cultural inventions: clothing on a giraffe will not allow it to survive in Antarctica, and neither shade nor shoes will help a salamander in the Sahara. I am not speaking of living in those places permanently, but simply as a measure of the durability of men under stress.
— Paul Shepard, The Tender Carnivore and the Sacred Game, 1973
Above is the only known film footage of Mark Twain, shot at Twain’s Connecticut home in 1909. The women are thought to be his daughters Clara and Jean.
These circles display an odd property — the three are linked, but no two are linked.
A.G. Smith exhibited this curious variant in Eureka in 1967:
“I leave it to the reader the problem of finding whether Knotung is knotted, and if so, whether it is equivalent to the Borromean Rings, with which it shares the property that cutting any one loop releases the other two completely.”
Samuel Johnson’s 1755 Dictionary of the English Language defines lizard as “an animal resembling a serpent, with legs added to it.”
Srinivasa Ramanujan devised this magic square to mark his own birthday. He began with a Latin square (upper right) in which the numbers 1, 2, 3, and 4 appear in each row, column, and long diagonal as well as in the four corners, the four central squares, the middle squares in the top and bottom rows, and the middle squares in the outermost columns. Note the adjustments that would be necessary to reduce the four top cells to zero, and arrange these adjustments in the diagonally reflected pattern shown in the upper left. Now adding these two squares together produces the square in the lower left, which gives us a formula for creating a magic square based on any date (in the format 1 January 2001). The example at lower right is based on Ramanujan’s own birthday, 22 December 1887 (so D = day = 22, M = month = 12, C = century = 18, and Y = year = 87). In this example all 16 numbers are distinct, but that won’t be the case with every date.
You and a friend agree to meet on New Year’s Day at the Mozart Café in Vienna. You fly separately to the city but are dismayed to learn that it contains multiple cafés by that name.
What now? On the first day each of you picks a café at random, but unfortunately you choose different locations. On the second day you could both go out searching cafés, but you might succeed only in “chasing each other’s tails.” On the other hand, if you both stay where you are, you’ll certainly never meet. What is your best course, assuming that you can’t communicate and that you must adopt the same strategy (with independent randomization)?
This distressingly familiar problem remains largely unsolved. If there are 2 cafés then the best course is to choose randomly between them each day. If there are 3 cafés, then it’s best to alternate between searching and staying put (guided by certain specified probabilities). But in cases of 4 or more cafés, the best strategy is unknown.
In 2007 a reader wrote to the Guardian, “I lost my wife in the crowd at Glastonbury. What is the best strategy for finding her?” Another replied, “Start talking to an attractive woman. Your wife will reappear almost immediately.”