Open and Shut

This pouch contains two counters. Each is either black or white. Without opening the pouch, prove that one is black and one white.

The probability that the pouch contains two black counters is 1/4; a black and white counter 1/2; and two white counters 1/4.

The chance of drawing one black counter in each case is 1, 1/2, or 0.

So if we combine these values we’ll get the probability of drawing one black counter from the pouch:

(1/4 × 1) + (1/2 × 1/2) + (1/4 × 0) = 1/2

And if the probability of drawing a black counter is 1/2, then the pouch must contain one black and one white counter. Q.E.D.

(J.A.H. Hunter, after Lewis Carroll)

A Premonition

A queer dream or illusion had haunted Lincoln at times through the winter [of 1860]. On the evening of his election he had thrown himself on one of the haircloth sofas at home, just after the first telegrams of November 6 had told him he was elected President, and looking into a bureau mirror across the room he saw himself full length, but with two faces. … A few days later he tried it once more and the illusion of the two faces again registered to his eyes. But that was the last; the ghost since then wouldn’t come back, he told his wife, who said it was a sign he would be elected to a second term, and the death pallor of one face meant he wouldn’t live through his second term.

— Carl Sandburg, Abraham Lincoln: The Prairie Years, 1926


When Dr. Franklin went to France on his revolutionary mission, his eminence as a philosopher, his venerable appearance, and the cause on which he was sent, rendered him extremely popular — for all ranks and conditions of men there entered warmly into the American interest. He was, therefore, feasted and invited to all the court parties. At these he sometimes met the old Duchess of Bourbon, who being a chess-player of about his force, they were very generally played together. Happening once to put her king into prise, the Doctor took it. ‘Ah,’ says she, ‘we do not take kings so.’ ‘We do in America,’ said the Doctor.

— Sarah Randolph, The Domestic Life of Thomas Jefferson, 1871

“Monkey and Pulley”

A rope is passed over a pulley. It has a weight at one end and a monkey at the other. There is the same length of rope on either side and equilibrium is maintained. The rope weighs four ounces per foot. The age of the monkey and the age of the monkey’s mother together total four years. The weight of the monkey is as many pounds as the monkey’s mother is years old. The monkey’s mother is twice as old as the monkey was when the monkey’s mother was half as old as the monkey will be when the monkey is three times as old as the monkey’s mother was when the monkey’s mother was three times as old as the monkey. The weight of the rope and the weight at the end is half as much again as the difference in weight between the weight of the weight and the weight and the weight of the monkey. Now, what is the length of the rope?

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“Almost Too Strange for Credence”

We all went over to Mannheim, and dined at the hotel where, seventeen years before, I, being fourteen months old, was given away to my aunt, who was also my godmother, to live with her forever as if I were her own child, and never to see my own parents, as such, any more. … When we returned to the station in the evening, we had a long time to wait for the train. On the platform was a poor woman, crying very bitterly, with a little child in her arms. Emmie Penrhyn, who was tender-hearted, went up to her, and said she was afraid she was in some great trouble. ‘Yes,’ she said, ‘it is about my little child. My little child, who is only fourteen months old, is going away from me forever in the train which is coming. It is going away to be adopted by its aunt, who is also its godmother, and I shall never, never have anything to do with it any more.’ It was of an adoption under exactly the same circumstances that we had been to Mannheim to keep the seventeenth anniversary!

— Augustus Hare, The Story of My Life, 1896