Four Play

It’s a popular recreation to try to arrange four 4s into various expressions to generate the whole numbers, like so:

1 = 4 ÷ 4 + 4 – 4
2 = 4 – (4 + 4) ÷ 4
3 = (4 × 4 – 4) ÷ 4
4 = 4 + 4 × (4 – 4)
5 = (4 × 4 + 4) ÷ 4

In 1881 a writer to the London journal Knowledge noted that each of the first 20 integers except 19 can be generated using the operations +, -, ×, and ÷. In 1964 Martin Gardner found that if you use square roots, decimals, factorials, concatenations (444), and overline (.444 …) then every positive integer less than 113 becomes possible. (113 is surprisingly hard; it becomes possible if you use percents or the gamma function.)

In 2001 a team of mathematicians from Harvey Mudd College found that you can even get four 4s to approximate some notable constants if you use a whip and a chair:

\displaystyle e \approx \left ( 4!! \right ) \sqrt[4!!]{\frac{\sqrt{4!!}}{4!!!}}

\displaystyle \pi \approx \sqrt{\sqrt{4!\cdot 4 + \sqrt{\sqrt{4\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{.4}}}}}}}}}} \approx 3.1415932

That expression for e is accurate to 21 decimal places; it can be made arbitrarily accurate by repeatedly replacing 4 with 4!. The authors note that similar expressions can be derived using three 3s or five 5s.

Amazingly, they also approximated g, the acceleration due to gravity, with four 4s, as well as Avogadro’s number NA.

(A. Bliss, S. Haas, J. Rouse and G. Thatte, “Math Bite: Four Constants in Four 4s,” Mathematics Magazine 74:4 [October 2001], 272.)

Some Enchanted Evening

In Southeast Asia, fireflies synchronize their flashing. Observing them in Siam in the 1920s, naturalist Hugh Smith wrote, “Imagine a tenth of a mile of river front with an unbroken line of [mangrove] trees with fireflies on every leaf flashing in synchronism. … Then, if one’s imagination is sufficiently vivid, he may form some conception of this amazing spectacle.”

The phenomenon was so unexpected that some initially dismissed the reports as an illusion; Phillip Laurent “could hardly believe [his] own eyes, for such a thing to occur among insects is certainly contrary to all natural laws.”

Each male fly’s flashes are initially sporadic, but they adjust their timing according to those around them until they’re synchronized. This helps identify them to females of their own species. Biologist John Buck observed, “Centers of synchrony built up slowly, two individuals often flashing independently for up to half a minute (about fifty cycles) before the flashes coincided. At this point their rhythms locked together and continued in synchrony thereafter.”

In 2015 Robin Meier and Andre Gwerder used LEDs to artificially direct the speed and rhythm of thousands of flashing fireflies (above), using this technique to “explore the idea of free will and transform a machine into a living actor inside a colony of insects.”

(Ying Zhou, Walter Gall, and Karen Nabb, “Synchronizing Fireflies,” College Mathematics Journal 37:3 [May 2006], 187-193.)

Podcast Episode 174: Cracking the Nazi Code

In 1940, Germany was sending vital telegrams through neutral Sweden using a sophisticated cipher, and it fell to mathematician Arne Beurling to make sense of the secret messages. In this week’s episode of the Futility Closet podcast we’ll describe the outcome, which has been called “one of the greatest accomplishments in the history of cryptography.”

We’ll also learn about mudlarking and puzzle over a chicken-killing Dane.

See full show notes …


Last year I mentioned that during Scotland’s 1904 Antarctic expedition, piper Gilbert Kerr had serenaded a penguin:

Well, by Ernest Shackleton’s expedition three years later they’d advanced to gramophones:

This doesn’t seem to have gone any better, but it’s increasingly clear that we’re the obtuse ones. Shackleton’s biologist, James Murray, wrote, “They came up to a party of strangers in a straggling procession, some big aldermanic fellow leading. At a respectful distance they halted, and the old male waddled close up and bowed gravely until his head almost touched his breast. With his head still bowed he made a long speech in a muttering manner, and having finished his speech he still kept his head bowed for a few seconds for politeness sake, and then raising it he described with his bill as large a circle as the joints of his neck would allow, and finally looked into our faces to see if we understood. If we had not, as usually was the case, he tried again.

“He was infinitely patient with our stupidity, but his followers were not so patient with him, and presently they would become sure that he was making a mess of it. Then another male would waddle forward and elbow the first Emperor aside as if to say, ‘I’ll show you how it ought to be done,’ and went again through the whole business.”

“They are the civilized nations of the Antarctic regions, and their civilization, if much simpler than ours, is in some respects higher and more worthy of the name.”


In a 2005 story about The Simpsons, San Francisco Chronicle writer Steve Rubenstein mentioned that in a dream Homer once “wrote that 1782 to the 12th power plus 1841 to the 12th power equals 1922 to the 12th power.” Rubenstein added, “(It does.)”

Well, it doesn’t — the first factor here must be even, and the second must be odd, so their sum can’t be even. The city desk prepared a correction saying that the equation was wrong, but Deputy Managing Editor Stephen R. Proctor pointed out that unless it gave the right answer, “the correction doesn’t correct.”

So they called in Sonoma State University mathematician Sam Brannen and produced this unusual notice:

A story Nov. 15 about mathematical references on “The Simpsons” TV show mistakenly said that 1,782 to the 12th power plus 1,841 to the 12th power equals 1,922 to the 12th power. Actually, 1,782 to the 12th power plus 1,841 to the 12th power equals 2,541,210,258,614, 589,176,288, 669, 958, 142, 428, 526,657, while 1,922 to the 12th power equals 2,541,210,259,314,801,410, 819, 278,649, 643,651,567,616.

Ombudsman Dick Rogers added, “Obviously.”


  • Dick Gregory gave his twin daughters the middle names Inte and Gration.
  • Trains were invented before bicycles.
  • “We must believe in free will — we have no choice.” — Isaac Bashevis Singer

“How Rumors Spread,” a palindrome by Fred Yannantuono:

“Idiot to idiot to idiot to idiot to idiot to idi …”

Southern Literature

south polar times 1

During Robert Falcon Scott’s first Antarctic expedition, 1901–04, Ernest Shackleton edited an illustrated magazine, the South Polar Times, to entertain the crew. Each issue consisted of a single typewritten copy that would circulate among up to 47 readers aboard the Discovery, Scott’s steam-powered barque, through each of two dark winters. Contributors would drop their anonymous essays, articles, and poems into a mahogany letterbox, and Shackleton composed each issue on a Remington typewriter perched atop a storeroom packing case.

The first issue appeared on April 23, 1902, and was, Shackleton noted, “greatly praised!” Scott wrote, “I can see again a row of heads bent over a fresh monthly number to scan the latest efforts of our artists, and I can hear the hearty laughter at the sallies of our humorists and the general chaff when some sly allusion found its way home. Memory recalls also the proud author expectant of the turn of the page that should reveal his work and the shy author desirous that his pages should be turned quickly.”

Shackleton was invalided home that summer, but other crewmembers took over the magazine for him that winter and indeed again on Scott’s second expedition in 1911. BBC History has some scans.

south polar times 2

(Anne Fadiman, “The World’s Most Southerly Periodical,” Harvard Review 43 [2012], 98-115.)

Flight Insurance

Again, speaking of probability, there is the story of the statistician who told a friend that he never takes airplanes. When asked why, he replied that he computed the probability that there be a bomb on the plane, and that although the probability was low, it was too high for his comfort.

A week later, the friend met him on a plane and asked him why he changed his theory. He replied: ‘I didn’t change my theory. It’s just that I subsequently computed the probability that there simultaneously be two bombs on the plane. This is low enough for my comfort, and so I now carry my own bomb.’

— Raymond Smullyan, A Mixed Bag, 2016