Another time-lengthening effect, the ‘watched-pot’ phenomenon, has been studied by Richard A. Block. Actually using the old adage ‘a watched pot never boils’ as the impetus for his experiments, Block tested the subjective time experiences of observers watching a pot of water as it was heated slowly to the boiling point. One group of subjects, told that they would subsequently be asked for a time estimate, attended carefully to the passage of time. They felt that the time interval was long. A second group, instructed that the experiment involved visual perception, attended to time less carefully and therefore estimated the duration to be shorter. One of Block’s conclusions is that attention to time has a strong influence on perceived length.

— Jonathan D. Kramer, The Time of Music, 1988

(Richard A. Block, Edward J. George, and Marjorie A. Reed, “A Watched Pot Sometimes Boils: A Study of Duration Experience,” Acta Psychologica 46:2 [1980], 81-94.)

From Reddit: Stare at the red dot on this woman’s nose for 30 seconds, then look at a white wall and blink.

Erasmus Darwin, 1786:

I was surprised, and agreeably amused, with the following experiment. I covered a paper about four inches square with yellow, and with a pen filled with a blue colour wrote upon the middle of it the word BANKS in capitals, and sitting with my back to the sun, fixed my eyes for a minute exactly on the centre of the letter N in the middle of the word; after closing my eyes, and shading them somewhat with my hand, the word was distinctly seen in the spectrum in yellow letters on a blue field; and then, on opening my eyes on a yellowish wall at twenty feet distance, the magnified name of BANKS appeared written on the wall in golden characters.

To interest his students in the nomenclature of organic chemistry, Hofstra University chemist Dennis Ryan designed compounds in the shapes of little figures. Shown here are oldmacdenynenynol, cowenynenynol, and turkenynenynol; he also designed a goose, a snake, a giraffe, and a duck.

See Small Business and A Little Story.

(Dennis Ryan, “Old MacDonald Named a Compound: Branched Enynenynols,” Journal of Chemical Education 74:7 [1997], 782.)

This relationship can be utilized as a trick by writing 12345679 and asking a person to select his favorite digit. Mentally multiply the digit he selected by 9, then write the result under the number above. Then say that inasmuch as he is fond of that digit he shall have plenty of it. Multiply the two numbers together and the digit he selected will result. Thus suppose 4 was selected; multiply 12345679 by 36, resulting in 444444444.

— Albert Beiler, Recreations in the Theory of Numbers, 1964

(I don’t know who came up with this — there’s a whole series of memes on this theme.)

If f(z) is a cubic polynomial with complex coefficients, and if the roots of f are three distinct non-collinear points A, B, and C in the complex plane, then the roots of the derivative f′ are the foci of the unique ellipse inscribed in triangle ABC and tangent to the sides at their midpoints.

The theorem is named for Morris Marden, but it had been proven about a century earlier by Jörg Siebeck.

(Dan Kalman, “The Most Marvelous Theorem in Mathematics,” Math Horizons 15:4 [April 2008], 16-17.)

There are only five integer-sided triangles in which the area equals the perimeter:

(5, 12, 13) (6, 8, 10) (6, 25, 29) (7, 15, 20) (9, 10, 17)

Only two of them are right triangles.

(Lee Markowitz, “Area = Perimeter,” Mathematics Teacher 74:3 [March 1981], 222-223.)