The Sleeping Beauty Problem

Prince Charming tells Sleeping Beauty, “I’m going to put you to sleep with this potion, and then I’ll flip a coin. Today is Sunday. If the coin lands heads, I’ll wake you again on Monday. If it lands tails, then I’ll wake you on Monday, put you to sleep again, and wake you on Tuesday. The potion induces a mild amnesia, so you won’t remember the intermediate awakening if it happens, but otherwise it won’t hurt you.”

When Sleeping Beauty awakes, what probability should she assign that the coin landed heads?

There seem to be two contradictory answers to this. From one perspective, the coin was fair, so it would seem the chance is 1/2. But from another, Beauty finds herself in one of three equally likely situations (heads/Monday, tails/Monday, and tails/Tuesday), so the chance of heads appears to be 1/3. Which is correct?


“My books are water; those of the great geniuses are wine. Everybody drinks water.” — Mark Twain

A Good Deed

Jim finds himself in the central square of a small South American town. Tied up against the wall are a row of twenty Indians, most terrified, a few defiant, in front of them several armed men in uniform. A heavy man in a sweat-stained khaki shirt turns out to be the captain in charge and, after a good deal of questioning of Jim which establishes that he got there by accident while on a botanical expedition, explains that the Indians are a random group of inhabitants who, after recent acts of protest against the government, are just about to be killed to remind other possible protestors of the advantages of not protesting. However, since Jim in an honoured visitor from another land, the captain is happy to offer him a guest’s privilege of killing one of the Indians himself. If Jim accepts, then as a special mark of the occasion, the other Indians will be let off. Of course, if Jim refuses, then there is no special occasion, and Pedro here will do what he was about to do when Jim arrived, and kill them all. Jim, with some desperate recollection of schoolboy fiction, wonders whether if he got hold of a gun, he could hold the captain, Pedro and the rest of the soldiers to threat, but it is quite clear from the set-up that nothing of that kind is going to work: any attempt at that sort of thing will mean that all the Indians will be killed, and himself. The men against the wall, and the other villagers, understand the situation, and are obviously begging him to accept. What should he do?

— Bernard Williams, “A Critique of Utilitarianism,” from Utilitarianism: For and Against, 1973


Which of these is Japanese sculptor Hananuma Masakichi, and which is the life-size wooden statue he completed in 1885? Amazingly, the statue’s on the right. Masakichi spent three years posing between two adjustable mirrors to capture every skin blemish, blue vein, and discoloration on his body, even inserting his own body hairs into hand-drilled holes at precise locations. He added glass eyes and eyelashes that were exact facsimiles of his own and applied a coat of lacquer to give the finished statue the appearance of flesh and blood. The finished product is so convincing that crowds reportedly had difficulty distinguishing the artist from his work when he posed next to it at exhibition. “The figure stands with a little mask in one hand and an instrument for carving in the other,” reported the Oriental Review. “The lifelike eyes are apparently gazing at the mask, and the face wears a look of intense absorption.”

Abbreviated Verse

A man hired by John Smith and Co.
Loudly declared that he’d tho.
Men that he saw
Dumping dirt near his store;
The drivers, therefore, didn’t do.

There is an old cook in N.Y.
Who insists you should always st.p.
Full vainly he’s tried
To eat some that was fried,
But he says he would rather ch.c.

The sermon our pastor Rt. Rev.
Began may have had a rt. clev.,
But his talk, though consistent,
Kept the end so far distant
That we left, since we felt he mt. nev.

— Anonymous

A Pretty Fact

three squares theorem

Given three adjacent squares,

a + b = c.

“Beauty is the first test,” wrote G.H. Hardy. “There is no permanent place in the world for ugly mathematics.”

“A Phonographic Cat”

This cat has a liking for the gramophone: it loves to get in the trumpet to sleep, and will not move even when a record is put in and played!

Strand, August 1906

The Gormanston Foxes

A curious legend attends the Viscounts Gormanston of Ireland. It is said that when the head of the house dies, the foxes leave the surrounding countryside and congregate at the door of the castle. The following statements, collected at the death of Jenico William Joseph, the 14th Viscount Gormanston, on Oct. 28, 1907, appeared in the New Ireland Review of April 1908:

  • Lady Gormanston: “At the time he was dying, foxes were seen about the house and coming towards the house for some days before. His valet who was sleeping in his room heard what he thought was a dog barking, and on going over to the window found that it was a fox sitting under the window and barking. … At the death of Edward, 13th Viscount, the foxes were also there. He had been rather better one day, but the foxes appeared, barking under the window, and he died that night contrary to expectation.”
  • Lucretia P. Farrell, daughter of the 13th Viscount: “On the day before my grandfather … died, the foxes came in pairs (an unusual thing) into the demesne from all the country round; they sat under his bedroom window, which was on the ground floor, and howled and barked all night, although constantly driven away only to return. … At my father’s death, in 1876, I had nursed him till the end, and before he died I fell ill, and my family told me that the foxes appeared in the same way, but not so many. … The Preston crest is a running fox, which we were told no other family has. This is all I can tell, and very creepy it was in those days.”
  • Anthony Delahan, coachman, and Patrick White, steward: “On Monday night, the 28th October, at about 8 o’clock, I saw two foxes in the chapel ground and five or six more round the front of the house and several more in the cloisters, which were circling round in a ring, crying all the time. I saw them continuously from then till about 11 o’clock when I went to bed. I took White with me who also saw the foxes.”
  • Richard Preston: “On Wednesday, 30th October, 1907, at about 10 p.m., I went down to the chapel at Gormanston Castle to watch by the remains of my father. … About 3 a.m. I first became consciously aware of a slight noise without the chapel. … [When I opened the side door,] sitting on the gravel path within four feet of where I stood was a full-grown fox. Just in the shadow, sitting close up against the walls of the chapel, was another. I could hear several more moving quietly about within a few yards, and was almost sure that I saw some of them. … Neither of the two which saw attempted to move until I left the chapel and took a step towards them. They then walked quietly off into the shadow.”

“We venture no comment on the evidence,” the editors write. “Our readers will appreciate it for themselves. Whatever be their interpretation of the facts, they will, at least, allow that in them there is something of the marvellous.”

The Magic Dice Cup

A tangram paradox from Sam Loyd’s Eighth Book of Tan (1903). Each of these cups was composed using the same seven geometric shapes. But the first cup is whole, and the others contain vacancies of different sizes.

“Of course it is a fallacy, a paradox, or an optical illusion, for you will say the feat is impossible!” But how is it done?

In a Word

n. the pleasure of anticipating victory or success