Parrondo’s Paradox

Imagine a staircase with 1001 stairs, numbered -500 to 500. You’re standing in the middle, on stair 0, and you want to reach the top. On each step you can play either of two coin-flipping games — if the result is heads then you move up a step; if it’s tails then you move down a step:

  • In game 1 you flip coin A, which is slightly biased: It comes up heads 49.5 percent of the time and tails 50.5 percent.
  • In game 2 you use two coins, B and C. Coin B produces heads 9.5 percent of the time and tails 90.5 percent. Coin C produces heads 74.5 percent of the time and tails 25.5 percent. In game 2 if the number of the stair you’re on is a multiple of 3 then you flip coin B; otherwise you flip coin C.

Both of these are losing games — if you played either game 1 or game 2 exclusively, you’d eventually find yourself at the bottom of the staircase. But in 1996 Spanish physicist Juan Parrondo found that if you play the two games in succession in random order, keeping your place on the staircase as you switch between them, you’ll rise to the top of the staircase. It’s not, properly speaking, a paradox, but it’s certainly counterintuitive.

This example is from David Darling’s Universal Book of Mathematics. (Thanks, Nick.)

“What’ll Be the Title?”

O to scuttle from the battle and to settle on an atoll far from brutal mortal neath a wattle portal!
To keep little mottled cattle and to whittle down one’s chattels and not hurtle after brittle yellow metal!
To listen, non-committal, to the anecdotal local tittle-tattle on a settle round the kettle,
Never startled by a rattle more than betel-nuts a-prattle or the myrtle-petals’ subtle throttled chortle!
But I’ll bet that what’ll happen if you footle round an atoll is you’ll get in rotten fettle living totally on turtle, nettles, cuttle-fish or beetles, victuals fatal to the natal élan-vital,
And hit the bottle.
I guess I’d settle
For somewhere ethical and practical like Bootle.

— Justin Richardson

Arms Race

Fly-plagued and enterprising in 1919, G.W. Blake came up with this inventive solution. The spring-loaded pistol shoots a projectile bearing a woven wire screen fast enough to surprise an unwitting fly who might have been expecting a low-tech flyswatter.

Next I suppose the flies will start shooting us.

A Disturbing Dinner

When British traveler Richard Gordon Smith reached Japan’s Lake Biwa in 1906, he asked a local cook whether he could prepare the rare dish koi-no-ikizukuri (“a carp cut up alive”), which had been offered at nobles’ feasts in ancient times. The cook, delighted, brought in a red lacquered tray decorated to resemble the sea bottom. On it lay a carp that opened and shut its mouth and gills as if it were swimming in water. “The dish was really pretty in spite of the gasping fish which, however, showed no pain, and there was not a sign of blood or a cut.” But “Now we are ready,” said the cook, and he dribbled some soy sauce into the fish’s eye:

The effect was not instantaneous: it took a full two minutes as the cook sat over him, chopsticks in hand. All of a sudden and to my unutterable astonishment, the fish gave a convulsive gasp, flicked its tail and flung the whole of its skin on one side of its body over, exposing the underneath of the stomach parts, skinned; the back was cut into pieces about an inch square and a quarter of an inch thick, ready for pulling out and eating.

“Never in my life have I seen a more barbarous or cruel thing,” Smith wrote, “not even the scenes at Spanish bull fights. Egawa [his interpreter] is a delicate-stomached person and as he could eat none, neither could I. It would be simply like taking bites out of a large live fish. I took the knife from my belt and immediately separated the fish’s neck vertebrae, much to the cook’s astonishment and perhaps disgust.” He asked them to take it away. “You have certainly operated beautifully,” he said, “but the sooner a law is brought in to prevent such cruelty the better.”

(From Travels in the Land of the Gods: The Japan Diaries of Richard Gordon Smith, 1986.)

The Jeweler’s Observation

Prove that every convex polyhedron has at least two faces with the same number of sides.

Click for Answer

Podcast Episode 59: The Wizard of Mauritius,_Ile_Maurice).jpg

In 1764 a French engineer on a tiny African island claimed that he could see ships beyond the horizon. In today’s show we’ll review the strange story of Étienne Bottineau and consider the evidence for his claims to have invented a new art.

We’ll also ponder a 400-year-old levitation trick and puzzle over why throwing a beer can at someone might merit a promotion.

Sources for our feature on nauscopie, the purported art of apprehending ships below the horizon:

Rupert T. Gould, Oddities: A Book of Unexplained Facts, 1928.

Sir David Brewster, Letters on Natural Magic, 1832.

J. Gregory Dill, “The Lost Art of Nauscopie,” Ocean Navigator, January/February 2003 (retrieved May 17, 2015).

Mike Dash, “Naval Gazing: The Enigma of Étienne Bottineau,” Smithsonian Magazine, Oct. 13, 2011 (retrieved May 17, 2015).

Chicago Tribune, “The Science of Nauscopie,” Nov. 7, 1869.

Greg’s post on Samuel Pepy’s “lifting experiment” appeared on Futility Closet on March 22, 2008. Further sources for that segment:

Sir David Brewster, Letters on Natural Magic, 1832.

The Diary of Samuel Pepys, July 31, 1665.

Robert Conger Pell, Milledulcia: A Thousand Pleasant Things Selected from “Notes and Queries,” 1857.

Notes & Queries, July 3, 1852 (the original query).

Notes & Queries, July 24, 1852 (Brewster offers his impressions).

“Non-Wist,” “Phenomenon of Levity in the Human Subject,” The Zoist, January 1852.

Two YouTube videos illustrate the modern technique: one, two

The YouTube discussion mentioned in this week’s lateral thinking puzzle is here (warning — this spoils the puzzle).

You can listen using the player above, download this episode directly, or subscribe on iTunes or via the RSS feed at

Please consider becoming a patron of Futility Closet — on our Patreon page you can pledge any amount per episode, and all contributions are greatly appreciated. You can change or cancel your pledge at any time, and we’ve set up some rewards to help thank you for your support.

You can also make a one-time donation via the Donate button in the sidebar of the Futility Closet website.

Many thanks to Doug Ross for the music in this episode.

If you have any questions or comments you can reach us at You can also follow us on Facebook and Twitter. Thanks for listening!


In illustrating his Just So Stories, Rudyard Kipling hid messages in the runic characters accompanying some drawings. The tusk above illustrates “How the First Letter Was Written”:

Left side: “This is the stori of Taffimai all ritten out on an old tusk. If u begin at the top left hand corner and go on to the right u can see for urself things as the happened.”

Right side: “The reason that I spell so queerli is becase there are not enough letters in the Runic alphabet for all the ourds that I ouant to use to u o beloved.”

Bottom (barely visible here): “This is the identical tusk on ouich the tale of Taffimai was ritten and etched bi the author.”

The initial “H” at the start of the “Cat That Walked by Himself” hides another message using the same characters: “I, Rudiard Kipling, drew this, but because there was no mutton bone in the house I faked the anatomi from memori.”

“Are these really Runic letters or just an alphabet that Kipling made up for fun?” asked Maj. B.J. Bewley in the Kipling Journal in January 1928. “I think the chief interest lies in the almost boyish pleasure the author plainly took in writing in these strange characters. He must have done it entirely for his own amusement.”

Double Duty

What’s unusual about this limerick?

There was a young lady of Riga,
Who went for a ride on a tiger,
They came back from their ride
With the lady inside
And a smile on the face of the tiger.

It remains a limerick when translated into Latin:

Puella Rigensis ridebat,
Quam tigris in tergo vehebat,
Externa profecta
Interna revecta,
Risusque cum tigre manebat.

Ronald Knox found that the same is true of this one:

There was a young man of Devizes,
Whose ears were of different sizes;
The one that was small
Was no use at all,
But the other won several prizes.

Visas erat; huic geminarum
Dispar modus auricularum:
Minor haec nihili;
Palma triplici
Iam fecerat altera clarum.

Asleep Awake

At age 13 Marie-Jean-Léon Lecoq, Marquis d’Hervey de Saint-Denys, discovered a rare talent: He could recognize a dream state while he was experiencing it, and could move and act lucidly within the dream. Eventually he filled 25 notebooks with descriptions and illustrations of his adventures in the dream world. These are now lost, but his 1867 book Les Rêves et les Moyens de Les Diriger describes some of his feats:

I change a porcelain vase into a rock-crystal fountain, from which I desire a cooling drink — and this immediately flows out through a golden tap. Some years ago I lost a particular ring whose loss I felt deeply. The memory of it comes to mind, and I should like to find it. I utter this wish, fixing my attention on a piece of coal that I pick up from the fireplace — and immediately the ring is on my finger. The dream continues in the same way until one of the apparitions I have called up charms and captivates me so much that I forget my magician’s role and plunge into a new, more realistic series of illusions.

Saint-Denys believed that almost anyone could learn to do this. One of his suggestions was to keep a dream diary and to make a daily habit of completing it. Like the rest of the student’s life, this habit would then itself become the raw material for his dreams — eventually he would dream of recording a dream. And if he noted the details of a dream he was recording, he would virtually be dreaming lucidly, having smuggled himself into his own slumbers.