The Digit Factory

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

Marden’s Theorem
Image: Wikimedia Commons

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.)

Applied Chemistry

On his May 1997 final exam at the University of Oklahoma School of Chemical Engineering, a Dr. Schlambaugh asked, “Is hell exothermic or endothermic? Support your answer with proof.” Most students based their responses on Boyle’s law, but one gave this answer:

First, we postulate that if souls exist, they must have some mass. If they do, then a mole of souls must have a mass. So at what rate are souls moving into hell and at what rate are souls leaving? I think we can safely assume that once a soul gets to hell it does not leave. Therefore, no souls are leaving. As for souls entering hell, let’s look at the different religions that exist in the world today. Some of the religions state that if you are not a member of their religion, you will go to hell. Since there are more than one of these religions and people do not belong to more than one religion, we can project that all souls go to hell. With the birth and death rates what they are, we can expect the number of souls in hell to increase exponentially. Now, we look at the rate of change in the volume of hell. Boyle’s Law states that in order for the temperature and pressure in hell to stay the same, the ratio of the mass of the souls to the volume needs to stay constant. (1) If hell is expanding at a slower rate than the rate at which souls enter hell, then the temperature and pressure in hell will increase until all hell breaks loose. (2) If hell is expanding at a rate faster than the increase in souls in hell, then the temperature and pressure will drop until hell freezes over. So which is it? If we accept the postulate given to me by Theresa Banyan during Freshman year, ‘It will be a cold night in hell before I sleep with you’ and take into account the fact that I still have not succeeded in having sexual relations with her, then (2) cannot be true. Thus hell is exothermic.

“The student, Tim Graham, got the only A.”

(Dave Morice, “Kickshaws,” Word Ways 31:2 [May 1998], 140-149.)

01/28/2020 This is a legend, apparently starting at the Taylor Instrument Company in the 1920s and accumulating some entertaining variations since then. The text of the Applied Optics piece is here. (Thanks, Dan and Pete.)

Midnight Oil

In trying to work out the trajectories of the planets, Johannes Kepler had to do reams of monstrous calculations by hand. In the manuscript pages for his revolutionary Astronomia nova of 1609, he concludes 15 folio pages of computations by writing:

“If thou art bored with this wearisome method of calculation, take pity on me, who had to go through with at least seventy repetitions of it, at a very great loss of time.”

Reversing Relations

The Book of Common Prayer includes a Table of Kindred and Affinity that lists prohibited degrees of marriage in the Church of England. For example, a man may not marry his daughter’s son’s wife, and a woman may not marry her husband’s mother’s father. In this case, the two proscriptions correspond — they describe the same relationship “from both sides,” so this union is prohibited to both parties in the relationship. But is this always the case? Is each union that’s denied to a man also denied to the woman? (The table lists only heterosexual unions.) It’s not immediately clear; 25 prohibited degrees are listed for each sex, and our language makes it hard to “reverse” the description of a relationship mentally.

In 1989 Manchester Polytechnic mathematician M.D. Stern worked out a notation that makes this easy. Use 1 to denote a male and 0 a female, and use this code to denote relationships between individuals:

00 spouse
01 parent
10 child
11 sibling

Now, to show the relationship between one person and another, write one digit for the first person followed by a sequence of three more digits — two to represent the relationship and one to represent the sex of the second person. So, taking the example above, a man’s daughter’s son’s wife would be denoted:

1 100 101 000

To interpret the same relationship from the woman’s point of view, we just reverse the order of the digits:

0 001 010 011

He is her husband’s mother’s father.

Applying this to the prohibited degrees in the table, Stern found that every prohibition for a man corresponds to an inverse prohibition for a woman — there are no prospective marriages that would be prohibited to one party but not the other.

(M.D. Stern, “A Notational Device for Analysing Relationships,” Mathematical Gazette 73:463 [March 1989], 37-40.)

More Prime Images

Inspired by James McKee’s Trinity Hall prime, physics researcher Gilles Esposito-Farese (of the self-descriptive pangram) has worked out that this 2,258-digit prime number:


renders these 7,500 digits in binary:

Esposito-Farese map

This is a 30,000-digit prime:

Esposito-Farese Gioconda

And this is self-explanatory:

Esposito-Farese prime declaration

More here.

(Thanks, Gilles.)

Nine Lives
Image: Wikimedia Commons
Image: Wikimedia Commons

In the 1960s, Soviet mathematician Vladimir Arnold mapped the square image of a cat to a torus, “stretched” (sheared) it as shown on that surface, then sliced the resulting image into pieces and recomposed them into a square.

As the process is repeated, any two points in the image quickly become separated, but, surprisingly, after sufficient repetitions the original image reappears.

A discrete analogue is at right. As the transformation is repeated, the image appears increasingly random or disordered, but the underlying cat can be glimpsed making occasional appearances, sometimes as a ghostly suggestion, sometimes in multiple smaller images, and occasionally (yowling, one imagines) even upside down.

It reappears again, unhurt, at the 300th iteration.

It’s called Arnold’s cat map. You can try it yourself here.

Advance Notice

In his 1966 book New Mathematical Diversions From Scientific American, Martin Gardner predicted that the millionth digit of π would be 5. (At the time the value was known only to about 10,000 decimal places.) He was reprinting a column on π from 1960 and included this in the addendum:

It will probably not be long until pi is known to a million decimals. In anticipation of this, Dr. Matrix, the famous numerologist, has sent me a letter asking that I put on record his prediction that the millionth digit of pi will be found to be 5. His calculation is based on the third book of the King James Bible, chapter 14, verse 16 (it mentions the number 7, and the seventh word has five letters), combined with some obscure calculation involving Euler’s constant and the transcendental number e.

He’d intended this as a hoax, but eight years later the computers discovered he was right.