The Longyou Caves
Image: Wikimedia Commons

In June 1992, farmers draining ponds in Longyou County, Quzhou prefecture, Zhejiang province, China, discovered that they weren’t ponds at all but drowned caverns, apparently created during the Ming Dynasty.

To date 36 such caves have been discovered in a region of 1 square kilometer. They’ve been compared to underground palaces, with rooms, halls, pillars, beds, bridges, and pavilions. But their age and function remain unclear because no historical document mentions them.

(Cheng Zhu et al., “Lichenometric Dating and the Nature of the Excavation of the Huashan Grottoes, East China,” Journal of Archaeological Science 40:5 [2013], 2485-2492.)

A Lucrative Loop

In 2014, Princeton mathematician John Conway proposed a simple conjecture:

Let n be a positive integer. Write the prime factorization in the usual way, e.g. 60 = 22 × 3 × 5, in which the primes are written in increasing order, and exponents of 1 are omitted. Then bring exponents down to the line and omit all multiplication signs, obtaining a number f(n). Now repeat.

So, for example, f(60) = f(22 × 3 × 5) = 2235. Next, because 2235 = 3 × 5 × 149, it maps, under f, to 35149, and since 35149 is prime, we stop there forever.

The conjecture, in which I seem to be the only believer, is that every number eventually climbs to a prime. The number 20 has not been verified to do so. Observe that 20 → 225 → 3252 → 223271 → …, eventually getting to more than one hundred digits without reaching a prime!

Conway offered $1,000 for a counterexample, a number that doesn’t “climb to a prime.” The challenge stood for three years before James Davis, who calls himself “not a mathematician by any stretch,” found the pretty 13532385396179 = 13 × 532 × 3853 × 96179, a composite number that loops onto itself and thus never reaches a prime — a find worth $1,000.

Red Flag Laws

In 1865, shortly after the first steam-powered horseless carriage appeared on English highways, Parliament ordered that a man must precede it on foot, carrying a red flag by day or a lantern by night, to warn others of the impending noise:

Firstly, at least three persons shall be employed to drive or conduct such locomotive, and if more than two waggons or carriages be attached thereto, an additional person shall be employed, who shall take charge of such waggons or carriages;
Secondly, one of such persons, while any locomotive is in motion, shall precede such locomotive on foot by not less than sixty yards, and shall carry a red flag constantly displayed, and shall warn the riders and drivers of horses of the approach of such locomotives, and shall signal the driver thereof when it shall be necessary to stop, and shall assist horses, and carriages drawn by horses, passing the same.

Vermont passed a similar law in 1894, requiring the owner of a steam-propelled vehicle to have a “person of mature age … at least one-eighth of a mile in advance of” the vehicle, to warn those with livestock of its approach. At night this person was required to carry a red light.

Both measures were repealed in 1896 — by which time the internal combustion engine was already being developed.

Reverse Buoyancy

It’s been established that a volume of fluid can be suspended stably above a layer of air by vibrating the whole system vertically. Now researchers in Paris have shown that lightweight objects can “float” on the underside of this suspended slab of liquid — thus inverting an entire seaside scene.

A YouTube commenter writes, “We’ve been sailing like that for years in Australia.”

(Benjamin Apffel et al., “Floating Under a Levitating Liquid,” Nature 585:7823 [2020], 48-52.)


In the climactic scene that Ray Harryhausen animated for Jason and the Argonauts (1963), “I had three men fighting seven skeletons, and each skeleton had five appendages to move in each separate frame of film. This meant at least thirty-five animation movements, each synchronized to the actor’s movements. Some days I was producing just 13 or 14 frames a day, or to put it another way, less than one second of screen time per day, and in the end the whole sequence took a record four and a half months to capture on film.”

An interesting philosophical question: “So how do you kill skeletons? We puzzled over this conundrum for some time and in the end we opted for simplicity by having Jason jump off the cliff into the sea, followed by the skeletons. It was the only way to kill off something that was already dead, and besides, we assumed that they couldn’t swim. After filming a stuntman jump into the sea, the prop men threw seven plaster skeletons off the cliff, which had to be done correctly on the first take as we couldn’t retrieve them for a second. To this day there are, somewhere in the sea near that hotel on the cliff edge, the plaster bones of seven skeletons.”

(Ray Harryhausen: An Animated Life, 2010.)

“Piccadilly Underground Station”

This unusual puzzle by G.A. Roberts appeared in the January 1941 issue of Eureka, the journal of recreational mathematics published at Cambridge University. It concerns the Piccadilly Circus station of the London Underground, which lies on the Piccadilly line between Green Park and Leicester Square and on the Bakerloo line between Charing Cross and Oxford Circus.

At a given time there are on the platform, escalators and subways, and in the trains, 128 people, all of whom travel by train, and none of whom return immediately by the way they have come.

Those who have come via Leicester Square are equal in number to those who are about to travel via Leicester Square.

The number of people who arrived by Bakerloo Line is equal to the number who intend to leave by the Piccadilly Line.

The number of people who are travelling from the street to stations on the Piccadilly Line is equal to six-thirteenths of the number who change from the Piccadilly Line to the Bakerloo.

The number who arrive from Green Park and then change to the Bakerloo is equal to the number who are about to travel via Green Park.

The number who are travelling from the street to the Bakerloo is equal to four times the number who arrive in Piccadilly trains but do not use the Bakerloo Line, and of these, twice as many come from Green Park as from Leicester Square.

By how many does the number of people who use the Bakerloo Line exceed that of those who do not?

Click for Answer

Engine Trouble

In John Milton’s 1637’s poem “Lycidas,” corrupt clergy are threatened with a obscure punishment:

The hungry Sheep look up, and are not fed,
But swoln with wind, and the rank mist they draw,
Rot inwardly, and foul contagion spread:
Besides what the grim Woolf with privy paw
Daily devours apace, and nothing sed,
But that two-handed engine at the door,
Stands ready to smite once, and smite no more.

What is the “two-handed engine”? That’s been a riddle for nearly 400 years. In 1950, Oberlin College philologist W. Arthur Turner collected 10 possibilities, ranging from the nations England and Scotland to “[t]he sheep-hook, which in Milton’s day apparently had an iron spud on the straight end and could be used as a weapon.” Turner himself thought that “the only engine which does meet all the requirements is the lock on St. Peter’s door (or the power of the lock), to which he carries the key.” But there’s still no strong consensus.

(W. Arthur Turner, “Milton’s Two-Handed Engine,” Journal of English and Germanic Philology 49:4 [October 1950], 562-565.)

“A Bully Is Always a Coward”

English proverbs:

  • Give neither counsel nor salt till you are asked for it.
  • A hedge between keeps friendship green.
  • A fault confessed is half redressed.
  • A hungry man is an angry man.
  • Please your eye and plague your heart.
  • If you run after two hares you will catch neither.
  • A good lawyer makes a bad neighbor.
  • Speak fair and think what you will.
  • It is not the suffering but the cause which makes a martyr.
  • A fool will laugh when he is drowning.
  • A foe is better than a dissembling friend.
  • A disease known is half cured.
  • Let your purse be your master.
  • Short counsel is good counsel.

And “Whosoever draws his sword against the prince must throw the scabbard away.”

Image: Wikimedia Commons

What does the middle initial “B.” stand for in Benoit B. Mandelbrot’s name?

Benoit B. Mandelbrot.

In the Notices of the American Mathematical Society, Andrew Kern calls this “My single favorite math joke of all.”

(Intriguingly, Mandelbrot adopted his middle initial; it does not stand for a middle name.)

03/02/2021 UPDATE: Reader Dan Uznanski sent another:

What’s an anagram for Banach-Tarski?

Banach-Tarski Banach-Tarski.