Letter from Albert Einstein to J.E. Switzer, April 23, 1953:

Dear Sir

Development of Western Science is based on two great achievements; the invention of the formal logical system (in Euclidean geometry) by the Greek philosophers, and the discovery of the possibility to find out causal relationship by systematic experiment (Renaissance). In my opinion one has not to be astonished that the Chinese sages have not made these steps. The astonishing thing is that these discoveries were made at all.

Sincerely yours,

A. Einstein

The “Lo Shu square” is the 3 × 3 square enclosed in dashed lines at the center of the diagram above. It’s “magic”: Each row, column, and long diagonal (marked in red) sums to 15. William Walkington has discovered a new magic property — imagine rolling the square into a tube (in either direction), and then bending the tube into a torus. And now imagine hopping from cell to cell around the torus with a “knight’s move” — two cells over and one up. (The extended diagram above helps with visualizing this — follow the blue lines.) It turns out that each such path touches three cells, and these cells always sum to 15. So the square is even more magic than we thought.

More info here. (Thanks, William.)

English chemist Septimus Piesse likened scents to music:

Odors seem to affect the olfactory nerves in certain definite degrees, as sounds act on the auditory nerves. There is, so to speak, an octave of smells, as there is an octave of tones; some perfumes accord, like the notes of an instrument. Thus almond, vanilla, heliotrope, and clematis, harmonize perfectly, each of them producing almost the same impression in a different degree. On the other hand, we have citron, lemon, orange peel, and verbena, forming a similarly associated octave of odors, in a higher key. The analogy is completed by those odors which we call half-scents, such as the rose, with rose-geranium for its semitone; ‘petit-grain’ and neroli, followed by orange-flower. With the aid of flowers already known, by mixing them in fixed proportions, we can obtain the perfume of almost all flowers.

Using an “odaphone,” or scale, on which harmonies and discords of odors might be studied, his London perfumery Piesse and Lubin produced some of the most important scents of the Victorian era, such as Ambergris (1873), Hungary Water (1873), Kiss Me Quick (1873), The Flower of the Day (1875), White Rose (1875), and Frangipanni (1880).

It had been thought that none of these had survived, but in 2011 two unopened bottles were discovered in the bow of the Mary Celestia, a Civil War blockade runner that had foundered off Bermuda in 1864. The bottles contained Bouquet Opoponax, one of the company’s most popular fragrances, and after analysis with a gas chromatograph, Germany’s Drom Fragrances managed to reproduce the scent in 2014.

“I was shocked at how fresh and floral it was and by the amount of citrus in it,” senior perfumer Jean-Claude Delville told the Star-Ledger. “When the fragrance has been sitting at the bottom of the ocean and aging for so many years you expect something that is oxidized or damaged,” he told CTVNews. “But my first impression was ‘wow’.”

This is interesting — if ax² + bx + c = 0, then

It’s called the citardauq formula — citardauq is quadratic spelled backward. It’s explained above by the inimitable James Tanton; here’s a Facebook discussion; and see the second answer here regarding questions of numerical instability.

Two climbers stand on opposite sides of a two-dimensional mountain range. Is it always possible for both of them to make their way through the mountains, remaining constantly at the same altitude as one another, and arrive together at the top of the tallest peak?

The example shown here looks relatively straightforward, but that doesn’t prove that it’s possible in every mountain range. Each time either climber reaches a peak or a valley, she must decide whether to go forward or back, and in a complex range it’s not always clear whether there’s a series of choices that will lead both climbers to the goal.

As it turns out, though, the answer is yes. Alan Tucker gave an accessible explanation, using graph theory, in the November 1995 issue of Math Horizons.

In 2003 Carl Libis of Assumption College in Worcester, Mass., received this solution from a student in an algebra course:

(Via Ed Barbeau, “Fallacies, Flaws, and Flimflam,” College Mathematics Journal 34:1 [January 2003], 50-54.)

Two years before he died, Lewis Carroll came up with a remarkable number-guessing game in which he’d send a volunteer through a long series of arithmetic manipulations, even allowing her to make some private decisions as to how to proceed. Along the way Carroll asked only three questions:

“Is the result odd or even?”

“Is the result odd or even?”

“How often does it go?”

And yet he would always be able to find the original number quickly. Here’s the procedure:

Think of a number (a positive integer).

Multiply by 3.

• If the result is odd, then add either 5 or 9 (whichever you like), then divide by 2, then add 1.
• If the result is even, then subtract either 2 or 6 (whichever you like), then divide by 2, then add 29 or 33 or 37 (whichever you like).

Multiply by 3.

• If the result is odd, then add either 5 or 9 (whichever you like), then divide by 2, then add 1.
• If the result is even, then subtract either 2 or 6 (whichever you like), then divide by 2, then add 29 or 33 or 37 (whichever you like).

Add 19 to original number you chose and append any digit, 0-9, to this number.

Add the previous result.

Divide by 7 and drop any remainder.

Divide by 7 again and drop any remainder, and tell me what result you get. (“How often does it go?”)

Here’s how to derive the number that was chosen originally:

Multiply the final answer by 4 and subtract 15. If the first answer was “even,” subtract 3 more, and if the second answer was “even,” subtract 2 more.

(Note: Carroll’s version contained an unfortunate flaw; the improvement given here was devised by Richard F. McCoart and includes a faster way to get the answer. See the article cited below. Carroll’s original appears in Morton N. Cohen’s Lewis Carroll, A Biography, 1995. Note too that the two “parity checks” above are identical, so the whole setup is easy to memorize and less bewilderingly complex than it’s intended to appear.)

(Richard F. McCoart, “Lewis Carroll’s Amazing Number-Guessing Game,” College Mathematics Journal 33:5 [November 2002], 378-383.)

This is remarkable — in 2014 ophthalmological neurologist Frederick Lepore showed this image to 100 migraine patients who experience a visual aura. (Click to enlarge.) 48 recognized it instantly.

“People are astonished,” Lepore told National Geographic. “They say, ‘Where did you get that?'”

He got it from English physician Hubert Airy, who had drawn his own aura in 1870, before the phenomenon was even understood.

(Via MetaFilter.)

The radius of a circle inscribed in a 3-4-5 triangle is 1.

(In fact, the inradius of any Pythagorean triangle is an integer.)