Illumination

http://commons.wikimedia.org/wiki/File:Karl_M%C3%BCller_Lesende_junge_Frau_beim_Licht_der_Petroleumlampe.jpg

Suppose I switch on my reading lamp at time zero. After one minute I switch it off again. Then I switch it on after a further 30 seconds, off after 15 seconds, and so on.

James Thomson asks: “At the end of two minutes, is the lamp on or off? … It cannot be on, because I did not ever turn it on without at once turning it off. It cannot be off, because I did in the first place turn it on, and thereafter I never turned it off without at once turning it on.”

What is the answer? Would the final state be different if I had switched the lamp off at time zero, rather than on? What if I carry out the experiment twice in succession?

See The Before-Effect.

Lost and Found

dalmatian photo

What is this? Most people see a mass of black blobs and then gradually recognize a photograph of a Dalmatian.

“What is interesting is that the outline shape on the picture surface that is experienced as resembling that of a dog is not seen as an outline shape at all unless the dog is seen in the figure,” writes University of British Columbia philosopher Dominic McIver Lopes. There’s no dog-shaped outline to notice; the contour of the dog’s body is invisible. To see the contour we must first see the dog … but how do we see the dog without the contour?

Ghost Writer

http://en.wikipedia.org/wiki/File:John-Cowper-Powys_2.jpg

[Theodore] Dreiser said that when he was living in New York, on West Fifty-seventh Street, John Cowper Powys came occasionally to dinner. At that time Powys was living in this country, in a little town about thirty miles up the Hudson, and he usually left Dreiser’s place fairly early to catch a train to take him home. One evening, after a rather long after-dinner conversation, Powys looked at his watch and said hurriedly that he had no idea it was so late, and he would have to go at once or miss his train. Dreiser helped him on with his overcoat, and Powys, on his way to the door, said, ‘ I’ll appear before you, right here, later this evening. You’ll see me.’

‘Are you going to turn yourself into a ghost, or have you a key to the door?’ Dreiser laughed when he asked that question, for he did not believe for an instant that Powys meant to be taken seriously.

‘I don’t know,’ said Powys. ‘ I may return as a spirit or in some other astral form.’

Dreiser said that there had been no discussion whatever during the evening, of spirits, ghosts or visions. The talk had been mainly about American publishers and their methods. He said that he gave no further thought to Powys’s promise to reappear, but he sat up reading for about two hours, all alone. Then he looked up from his book and saw Powys standing in the doorway between the entrance hall and the living room. The apparation had Powys’s features, his tall stature, loose tweed garments and general appearance, but a pale white glow shone from the figure. Dreiser rose at once, and strode towards the ghost, or whatever it was, saying, ‘Well, you’ve kept your word, John. You’re here. Come on in and tell me how you did it.’ The apparation did not reply, and it vanished when Dreiser was within three feet of it.

As soon as he had recovered somewhat from his astonishment Dreiser picked up the telephone and called John Cowper Powy’s house in the country. Powys came to the phone, and Dreiser recognized his voice. After he had heard the story of the apparation, Powys said, ‘I told you I’d be there, and you oughtn’t to be surprised.’ Dreiser told me that he was never able to get any explanation from Powys, who refused to discuss the matter from any standpoint.

— W.E. Woodward, The Gift of Life, 1947

Double and Half

A “curious paradox” presented by Raymond Smullyan at the first Gathering for Gardner: Consider two positive integers, x and y. One is twice as great as the other, but we’re not told which is which.

  • If x is greater than y, then x = 2y and the excess of x over y is equal to y. On the other hand, if y is greater than x, then x = 0.5y and the excess of y over x is y – 0.5y = 0.5y. Since y is greater than 0.5y, then we can say generally that the excess of x over y, if x is greater than y, is greater than the excess of y over x, if y is greater than x.
  • Let d be the difference between x and y. This is the same as saying that it’s equal to the lesser of the two. Generally, then, the excess of x over y, if x is greater than y, is equal to the excess of y over x, if y is greater than x.

The two conclusions contradict one another, so something is amiss. But what?

Close to Home

http://commons.wikimedia.org/wiki/File:FujitaNobuo.jpg

On Sept. 9, 1942, a lookout on Mount Emily in Oregon’s Siskiyou National Forest reported a plume of smoke near the town of Brookings. The Forest Service contained the fire easily, but investigators turned up something odd at the site: fragments of an incendiary bomb of Japanese origin.

It turned out that a Japanese submarine had surfaced off the Oregon/California border and 31-year-old navy officer Nobuo Fujita had piloted a seaplane into the forest, hoping to start a fire that would divert U.S. military resources from the Pacific. Recent rains had wet the forest, so the plan failed, but it marked the first time the continental United States had been bombed by enemy aircraft.

Fujita returned safely to Japan, where he opened a hardware store after the war, and he became an agent of amity with the United States. In 1962 he accepted an invitation to return to Oregon, where he donated his family’s samurai sword to Brookings, and he invited three local students to visit Japan in 1985. The city made him an honorary citizen shortly before his death in 1997, and his daughter spread his ashes at the site of the bombing.

Sand Reckoning

Using only a 4-minute hourglass and a 7-minute hourglass, how can you measure 9 minutes?

Click for Answer

Boor Laws

http://commons.wikimedia.org/wiki/File:Chamberlin_-_Benjamin_Franklin_(1762).jpg

Ben Franklin’s “rules for making oneself a disagreeable companion,” 1750:

  1. If possible engross the whole Discourse; and when other Matter fails, talk much of your-self, your Education, your Knowledge, your Circumstances, your Successes in Business, your Victories in Disputes, your own wise Sayings and Observations on particular Occasions, &c. &c. &c.;
  2. If when you are out of Breath, one of the Company should seize the Opportunity of saying something; watch his Words, and, if possible, find somewhat either in his Sentiment or Expression, immediately to contradict and raise a Dispute upon. Rather than fail, criticise even his Grammar.
  3. If another should be saying an indisputably good Thing; either give no Attention to it; or interrupt him; or draw away the Attention of others; or, if you can guess what he would be at, be quick and say it before him; or, if he gets it said, and you perceive the Company pleas’d with it, own it to be a good Thing, and withal remark that it had been said by Bacon, Locke, Bayle, or some other eminent Writer; thus you deprive him of the Reputation he might have gain’d by it, and gain some yourself, as you hereby show your great Reading and Memory.
  4. When modest Men have been thus treated by you a few times, they will chuse ever after to be silent in your Company; then you may shine on without Fear of a Rival; rallying them at the same time for their Dullness, which will be to you a new Fund of Wit.

“Thus you will be sure to please yourself,” he concluded. “The polite Man aims at pleasing others, but you shall go beyond him even in that. A Man can be present only in one Company, but may at the same time be absent in twenty. He can please only where he is, you where-ever you are not.”

Spin Control

http://www.sxc.hu/photo/443691

We’re playing Russian roulette. The revolver has six chambers, all empty. I put bullets in two adjacent chambers, spin the cylinder, hold the gun to my head, and pull the trigger. It clicks. Now it’s your turn. Before pulling the trigger, you can choose to spin the cylinder again or leave it as it is. Which is better?

Click for Answer

“A Magic Circle of Cubes”

kaprekar circle of cubes

Reading this circle clockwise produces the numbers 04, 20, 34, 12, 50, 42, 03, 41, 53, 15, 31, 25.

Reading it counterclockwise gives 05, 21, 35, 13, 51, 43, 02, 40, 52, 14, 30, 24.

The sum of the first group equals that of the second, and this holds true if the numbers are squared or cubed. Further, if the numbers in the first group are arranged in ascending order and those in the second in descending, then:

kaprekar circle sums

(Devised by D.R. Kaprekar in 1956.)