Each number in this pandiagonal order-4 magic square is a three-digit prime:

277 197 631 431
661 401 307 167
137 337 491 571
461 601 107 367

Add 30 to each cell and you get a new magic square, also made up of 16 three-digit primes:

307 227 661 461
691 431 337 197
167 367 521 601
491 631 137 397

Add 1092 to each cell in that one and you get a magic square of four-digit primes:

1399 1319 1753 1553
1783 1523 1429 1289
1259 1459 1613 1693
1583 1723 1229 1489

(Allan William Johnson Jr., “Related Magic Squares,” Journal of Recreational Mathematics 20:1 [January 1988], 26-27, via Clifford A. Pickover, The Zen of Magic Squares, Circles, and Stars, 2011.)

The Chinese mathematician Liu Hui offered this technique in a text composed about 500 years after Euclid. We’re on the mainland, and we want to find the height of a mountain on a distant island without crossing the sea.

Liu Hui showed that this can be accomplished by setting up two poles of a known height in a line with the mountain …

… and by appealing to a principle of complementary rectangles — here the red and the blue rectangles have the same area:

By using that principle it’s possible to recast the problem in terms of values that we can measure: the height of the poles (CD), the “offset” from which the top of the mountain can just be sighted from ground level over the top of each pole (DG and FH), and the distance between the poles (DF). Putting all that together we can find both the height of the mountain:

and the distance between the first pole and the mountain:

without ever leaving the mainland. Penn State University mathematician Frank Swetz concluded that “in the endeavours of mathematical surveying, China’s accomplishments exceeded those realized in the West by about one thousand years.”

“There is nothing at McDonald’s that makes it necessary to have teeth.” — Harvard nutritionist Jean Mayer, Time, 1973

Any two lines drawn between opposite sides of a square and intersecting at right angles are equal.

The quicksort computer sorting algorithm demonstrated with Hungarian folk dance, from Romania’s Sapientia University.

Also:

The four queens puzzle solved using ballet.

Binary search through flamenco dance.

Merge sort via Transylvanian-Saxon folk dance.

Selection sort using Gypsy folk dance.

More.

(Via MetaFilter.)

01/19/2019 UPDATE: When Gavin Taylor showed these algorithms to his students at the United States Naval Academy, they asked whether they themselves could dance for extra credit. He said yes. So here are the U.S. Naval Academy midshipmen dancing the InsertionSort algorithm:

(Thanks, Gavin.)

Richard Feynman was so struck by this fact that he remembered ever afterward where he had learned it — from his childhood friend Morrie Jacobs as the two stood in Morrie’s father’s leather shop in Far Rockaway, Queens.

If you know the vertices of a polygon, here’s an interesting way to find its area:

1. Arrange the vertices in a vertical list, repeating the first vertex at the end (see below).
2. Multiply diagonally downward both ways as shown.
3. Add the products on each side.
4. Find the difference of these sums.
5. Halve that difference to get the area.

This works for any polygon, no matter the number of points, so long as it doesn’t intersect itself. It’s a slight restatement of the shoelace formula.

(Thanks, Derek, Dan, and Kyle.)

A.F. Bainbridge of British Aerospace noticed this curiosity in 1991. On a calculator keypad like this:

1 2 3
4 5 6
7 8 9

… choose two three-digit numbers (say, 435 and 667) and multiply them (290145). Now use symmetrical paths on the keyboard to find two “complementary” numbers (that is, symmetrical across the center, here 675 and 443) and multiply those (299025).

The difference between these two products (299025 – 290145 = 8880) will always be evenly divisible by 37.

(A.F. Bainbridge and P.A. Binding, “Symmetrical Paths on a Calculator,” Mathematical Gazette 75:474 [December 1991], 399-401.)