A Guest Appearance

The Fibonacci numbers are the ones in this sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

Each number is the sum of the two that precede it. But now, interestingly:

\displaystyle  \mathrm{arctan} \left ( \frac{1}{1} \right ) = \mathrm{arctan} \left ( \frac{1}{2} \right ) + \mathrm{arctan} \left ( \frac{1}{3} \right )\\  \mathrm{arctan} \left ( \frac{1}{3} \right ) = \mathrm{arctan} \left ( \frac{1}{5} \right ) + \mathrm{arctan} \left ( \frac{1}{8} \right )\\  \mathrm{arctan} \left ( \frac{1}{8} \right ) = \mathrm{arctan} \left ( \frac{1}{13} \right ) + \mathrm{arctan} \left ( \frac{1}{21} \right )\\  \mathrm{arctan} \left ( \frac{1}{21} \right ) = \mathrm{arctan} \left ( \frac{1}{34} \right ) + \mathrm{arctan} \left ( \frac{1}{55} \right )\\

“And so on!” writes James Tanton in Mathematics Galore! (2012). “The first relation, for instance, states that a line of slope 1/2 stacked with a line of slope 1/3 gives a line of slope 1. (Can you prove the relations?)”

(Ko Hayashi, “Fibonacci Numbers and the Arctangent Function,” Mathematics Magazine 76:3 [June 2003], 215.)

Vulture Picnic

For her 2009 work In Ictu Oculi (“In the Twinkling of an Eye”), artist Greta Alfaro spread a table outside the Spanish village of Fitero and filmed a feast among 40 vultures.

“It was not easy to get them to jump on the table,” she told the Translocal Institute for Contemporary Art. “I had to wait for one week, setting the table every morning and unsetting it at dusk. Vultures have extraordinary eyesight, and if one of them notices that there is food, it will draw circles in the air to let the others know. They approached the scene every day, but either my presence or the presence of the table prevented them from getting closer.”

“I think that it is important today to reflect on the impermanence of almost everything, and on the fact that life cannot be controlled.”

No Waiting

In 1892 … a law firm in the American West came up with the idea of a divorce papers vending machine. For a while, at least, legal divorce papers were items that could be bought from a vending machine in Corinne, Utah. A purchaser could insert $2.50 in coins, pull a lever on the side of the machine, and pick up his papers from a delivery drawer that popped open like a cash register drawer. Those papers were then taken to the local law firm — whose name was printed on the form — where the names of the divorcing couple were written in and witnessed.

— Kerry Segrave, Vending Machines: An American Social History, 2002



“Stopping by Euclid’s Proof of the Infinitude of Primes,” by Presbyterian College mathematician Brian D. Beasley, “with apologies to Robert Frost”:

Whose proof this is I think I know.
I can’t improve upon it, though;
You will not see me trying here
To offer up a better show.

His demonstration is quite clear:
For contradiction, take the mere
n primes (no more), then multiply;
Add one to that … the end is near.

In vain one seeks a prime to try
To split this number — thus, a lie!
The first assumption was a leap;
Instead, the primes will reach the sky.

This proof is lovely, sharp, and deep,
But I have promises to keep,
And tests to grade before I sleep,
And tests to grade before I sleep.

(From Mathematics Magazine 78:2 [April 2005], 171.)

Notes and Measures

Howard Shapiro chose an unusual way to present his paper “Fluorescent Dyes for Differential Counts by Flow Cytometry” at the 1977 meeting of the Histochemical Society — he sang it:

Blood cells are classified by cell and nuclear shape and size
And texture, and affinity for different types of dyes,
And almost all of these parameters can quickly be
Precisely measured by techniques of flow cytometry.

It’s hard to fix a cell suspension rapidly and stain
With several fluorochromes, and this procedure, while it plain-
Ly furnishes the data which one needs to classify,
May fade away, and newer, simpler, methods never dye. …

The full paper, 76 verses with figures and sheet music, is here.

Dead Issues


In his early handwritten notes for Dracula, Bram Stoker considered giving the vampire these attributes:

  • can banish good thoughts, create evil thoughts and destroy will
  • is affected only by relics that are older than he is
  • cannot be painted, any portrait looks like someone else
  • cannot be photographed, photographs come out black or like a skeleton corpse
  • insensitive to music
  • cannot cross thresholds without assistance, stumbles on threshold
  • can determine and prove if people are sane
  • leeches are attracted to him, then repulsed
  • can pick out murderers
  • despises death and the dead
  • can tell if bodies are dead or alive

Jonathan Harker’s stay at Castle Dracula was originally to include “an encounter with a ‘wehr wolf’,” and at the London zoological gardens “Dracula enrages eagles and lions but intimidates wolves and hyenas.”

The notes also shed some light on a puzzle I’d mentioned earlier: Why does Dracula choose England in the first place? Stoker’s notes include the phrase “English law directory sortes Virgilianae central place marked with point of knife.” Sortes Virgilianae is Latin for Virgilian lots, a form of divination in which advice or predictions are sought by interpreting passages from Virgil. In Bram Stoker’s Notes for Dracula, Robert Eighteen-Bisang writes, “Did Dracula choose his law firm by a stabbing a knife into a law directory, or decide on the location of his new home by thrusting a knife into a map? The vampire’s use of divination is in keeping with the supposition that he is a sorcerer.” None of this made it into the final novel, but it might still be the explanation that Stoker had in mind.

The Fifth Card


I hand you an ordinary deck of 52 cards. You inspect and shuffle it, then choose five cards from the deck and hand them to my assistant. She looks at them and passes four of them to me. I name the fifth card.

At first this appears impossible. The hidden card is one of 48 possibilities, and by passing me four cards in some order my assistant can have sent me only 1 of 4! = 24 messages. How am I able to name the card?

Part of the secret is that my assistant gets to choose which card to withhold. The group of five cards that you’ve chosen must contain two cards of the same suit. My assistant chooses one of these to be the hidden card and passes me the other one. Now I know the suit of the hidden card, and there are 12 possibilities as to its rank. But my assistant can pass me only three more cards, with 3! = 6 possible messages, so the task still appears impossible.

The rest of the secret lies in my assistant’s choice as to which of the two same-suit cards to give me. Think of the 13 card ranks arranged in a circle (with A=1, J=11, Q=12, and K=13). Given two ranks, it’s always possible to get from one to the other in at most 6 steps by traveling “the short way” around the circle. So we agree on a convention beforehand: We’ll imagine that the ranks increase in value A-K, and the suits as in bridge (or alphabetical) order, clubs-diamonds-hearts-spades. This puts the whole deck into a specified order, and my assistant can pass me the three remaining cards in one of six ways:

{low, middle, high} = 1
{low, high, middle} = 2
{middle, low, high} = 3
{middle, high, low} = 4
{high, low, middle} = 5
{high, middle, low} = 6

So if my assistant knows that I’ll always travel clockwise around the imaginary circle, she can choose the first card to establish the suit of the hidden card and to specify one point on the circle, and then order the remaining three cards to tell me how many clockwise steps to take from that point to reach the hidden rank.

“If you haven’t seen this trick before, the effect really is remarkable; reading it in print does not do it justice,” writes mathematician Michael Kleber. “I am forever indebted to a graduate student in one audience who blurted out ‘No way!’ just before I named the hidden card.”

It first appeared in print in Wallace Lee’s 1950 book Math Miracles. Lee attributes it to William Fitch Cheney, a San Francisco magician and the holder of the first math Ph.D. ever awarded by MIT.

(Michael Kleber, “The Best Card Trick,” Mathematical Intelligencer 24:1 [December 2002], 9-11.)


In introducing the puzzle-loving logician Raymond Smullyan, the chairman of a meeting praised him as unique.

“I’m sorry to interrupt you, sir,” said Smullyan, “but I happen to be the only one in the entire universe who is not unique.”

Podcast Episode 107: Arthur Nash and the Golden Rule


In 1919, Ohio businessman Arthur Nash decided to run his clothing factory according to the Golden Rule and treat his workers the way he’d want to be treated himself. In this week’s episode of the Futility Closet podcast we’ll visit Nash’s “Golden Rule Factory” and learn the results of his innovative social experiment.

We’ll also marvel at metabolism and puzzle over the secrets of Chicago pickpockets.

See full show notes …

Risky Business

Image: Wikimedia Commons

In 1982, MIT physicist A.P. French received this letter from a writer in New Rochelle, N.Y.:

Being a safety minded individual I thought I would write you before experimenting on my own. Is it safe to mix Antipasto and Pasta together and could this be a future energy supply?

He responded:

I believe that your thoughtful and interesting suggestion about the mixing of pasta and antipasto deserves some acknowledgment. This process might well be a significant energy source — but only, I think, intragastrically. I estimate that the digestion of 1 lb of the mixture would release energy equivalent to about 0.001 megawatt hours or 0.000001 kilotons of TNT. I would not foresee any unusual hazards.

(From Robert L. Weber, ed., Science With a Smile, 1992.)