An anonymous proof that heaven is hotter than hell, from Applied Optics, August 1972:
The temperature of Heaven can be rather accurately computed from available data. Our authority is the Bible: Isaiah 30:26 reads, Moreover the light of the Moon shall be as the light of the Sun and the light of the Sun shall be sevenfold, as the light of seven days. Thus Heaven receives from the Moon as much radiation as the Earth does from the Sun and in addition seven times seven (forty-nine) times as much as the Earth does from the Sun, or fifty times in all. The light we receive from the Moon is a ten-thousandth of the light we receive from the Sun, so we can ignore that. With these data we can compute the temperature of Heaven: The radiation falling on Heaven will heat it to the point where the heat lost by radiation is just equal to the heat received by radiation. In other words, Heaven loses fifty times as much heat as the Earth by radiation. Using the Stefan-Boltzmann fourth-power law for radiation
where E is the absolute temperature of the Earth — 300K. This gives H as 798K absolute (525°C).
The exact temperature of Hell cannot be computed but it must be less than 444.6°C, the temperature at which brimstone or sulfur changes from a liquid to a gas. Revelations 21:8: But the fearful and unbelieving … shall have their part in the lake which burneth with fire and brimstone. A lake of molten brimstone means that its temperature must be below the boiling point, which is 444.6°C. (Above that point it would be a vapor, not a lake.)
We have then, temperature of Heaven, 525°C. Temperature of Hell, less than 445°C. Therefore, Heaven is hotter than Hell.
While a student at Cambridge, Paul Dirac attended a mathematical congress that posed the following problem:
After a big day’s catch, three fisherman go to sleep next to their pile of fish. During the night, one fisherman decides to go home. He divides the fish in three and finds that this leaves one extra fish. He throws this into the water, takes one third of the remaining fish, and departs.
The second fisherman awakes. Not knowing that the first has left, he too divides the fish into three piles, finds one fish left over, discards it, and takes a third of the remainder. The third fisherman does the same. What is the least number of fish that the fishermen could have started with?
Dirac proposed that they had begun with -2 fish. The first fisherman threw one into the water, leaving -3, and took a third of this, leaving -2. The second and third fisherman followed suit.
This story was recalled by “a well-meaning experimenter” in the Russian miscellany Physicists Continue to Laugh (1968). “I could tell many other stories about theoreticians and their work,” he wrote, “but they have told me that one theoretician is writing a story under the title ‘How Experimental Physicists Work.’ That, of course, will be presented upside down.”
In 1948, George Washington University doctoral student Ralph Alpher was working on a cosmology thesis under physicist George Gamow. As the paper took shape, “Gamow, with the usual twinkle in his eye, suggested that we add the name of Hans Bethe to an Alpher-Gamow letter to the editor of the Physical Review,” listing the authors as Alpher-Bethe-Gamow.
Bethe agreed to join, and the result, now known as the αβγ paper, was published on April 1, 1948 (“believe it or not, a date not of our asking”). “The response was fascinating,” Alpher later recalled, “ranging from feature articles, Sunday supplement stories, newspaper cartoons and voluminous mail from religious fundamentalists, to a packed audience of over 200, including members of the press, at the traditionally public (though usually not in this sense) ‘defence’ of the thesis.”
Gamow added, “There was, however, a rumor that later, when the alpha, beta, gamma theory went temporarily on the rocks, Dr. Bethe seriously considered changing his name to Zacharias.”
How to tell a parrot from a carrot, from American physicist Robert W. Wood’s extracurricular How to Tell the Birds From the Flowers: A Manual of Flornithology for Beginners (1907):
The Parrot and the Carrot we may easily confound,
They’re very much alike in looks and similar in sound.
We recognize the Parrot by his clear articulation,
For Carrots are unable to engage in conversation.
Below: A further distinction.
“Standards for inconsequential trivia,” offered by Philip A. Simpson in the NBS Standard, Jan. 1, 1970:
10-15 bismols = 1 femto-bismol
10-12 boos = 1 picoboo
1 boo2 = 1 boo-boo
10-18 boys = 1 attoboy
1012 bulls = 1 terabull
101 cards = 1 decacards
10-9 goats = 1 nanogoat
2 gorics = 1 paregoric
10-3 ink machines = 1 millink machine
109 los = 1 gigalos
10-1 mate = 1 decimate
10-2 mentals = 1 centimental
10-2 pedes = 1 centipede
106 phones = 1 megaphone
10-6 phones = 1 microphone
1012 pins = 1 terapin
A puzzle from the Middle Ages, adapted by A.N. Prior:
Four people, on a certain occasion, say one thing each.
A says that 1 + 1 = 2.
B says that 2 + 2 = 4.
C says that 2 + 2 = 5.
Can D now say that exactly as many truths as falsehoods are uttered on this occasion?
“If what D says is true,” Prior writes, “that makes 3 truths to 1 falsehood, so that it is false; while if it is false, that makes two truths and two falsehoods, and it is true.”
Choose a prime number p, draw a p×p array, and fill it with integers like so:
Now: Can we always find p cells that contain prime numbers such that no two occupy the same row or column? (This is somewhat like arranging rooks on a chessboard so that every rank and file is occupied but no rook attacks another.)
The example above shows one solution for p=11. Does a solution exist for every prime number? No one knows.
What is the smallest integer that’s not named on this blog?
Suppose that the smallest integer that’s not named (explicitly or by reference) elsewhere on the blog is 257. But now the phrase above refers to that number. And that instantly means that it doesn’t refer to 257, but presumably to 258.
But if it refers to 258 then actually it refers to 257 again. “If it ‘names’ 257 it doesn’t, so it doesn’t,” writes J.L. Mackie, “but if it doesn’t, then it does, so it does.”
(Adduced by Max Black of Cornell.)
Mr. X, who thinks Mr. Y a complete idiot, walks along a corridor with Mr. Y just before 6 p.m. on a certain evening, and they separate into two adjacent rooms. Mr. X thinks that Mr. Y has gone into Room 7 and himself into Room 8, but owing to some piece of absent-mindedness Mr. Y has in fact entered Room 6 and Mr. X Room 7. Alone in Room 7 just before 6, Mr. X thinks of Mr. Y in Room 7 and of Mr. Y‘s idiocy, and at precisely 6 o’clock reflects that nothing that is thought by anyone in Room 7 at 6 o’clock is actually the case. But it has been rigorously proved, using only the most general and certain principles of logic, that under the circumstances supposed Mr. X just cannot be thinking anything of the sort.
– A.N. Prior, “On a Family of Paradoxes,” Notre Dame Journal of Formal Logic, 1961
Dick and Jane are playing a game. Each holds up one or two fingers. If the total number of fingers is odd, then Dick pays Jane that number of dollars. If it’s even, then Jane pays Dick:
At first blush this looks fair, but in fact it’s distinctly favorable for Jane. Let p be the proportion of times that Jane holds up one finger. Her average winnings when Dick holds up one finger are -2p + 3(1 – p), and her average winnings when he holds up two fingers are 3p – 4(1 – p). If she sets those equal to one another she gets p = 7/12. This means that if she raises one finger with probability 7/12, then on average she’ll win -2(7/12) + 3(5/12) = 1/12 dollar every round, no matter what Dick does. Dick’s best strategy is also to raise one finger 7/12 of the time, but the best this can do is to restrict his loss to 1/12 dollar on average. It’s not a fair game.
Why do mathematicians confuse Halloween with Christmas?
Because 31 Oct = 25 Dec.
In 1976, Queen Elizabeth College chemist Leslie Hough asked graduate researcher Shashikant Phadnis to test a certain chlorinated sugar compound. Phadnis, whose English was imperfect, “thought I needed to taste it! … So I took a small quantity of the sample on a spatula and tasted it with the tip of my tongue.”
To his surprise, Phadnis found the compound intensely and pleasantly sweet. When he reported his discovery to Hough, “‘Are you crazy or what?’ he asked me. ‘How could you taste compounds without knowing anything about their toxicity?’” After some further cautious tasting, Hough dubbed the compound Serendipitose. It became the artificial sweetener Splenda.
“Later on, Les even had a cup of coffee sweetened with a few particles of Serendipitose. When I reminded him that it could be toxic (as it contained a high proportion of chlorine), he simply said, ‘Oh, forget it, we’ll survive!’”
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.
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?
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:
(Devised by D.R. Kaprekar in 1956.)
Illinois State University mathematician Phil Grizzard points out that a person born on Nov. 30, 1999, is a “stopwatch baby” — the date always displays her age in months, days, and years. For example, today, 5/4/12, such a person has been alive for 5 months, 4 days, and 12 years. (Europeans can swap the month and day — the principle still works.)
A caveat: In December we must “make change” by setting the month to 0 and adding 1 to the year. So this Christmas, 12/25/12, a stopwatch baby will be 0/25/13 — 0 months, 25 days, and 13 years old.
How to get rich using pocket handkerchiefs, from Lewis Carroll’s Sylvie and Bruno Concluded:
Here Lady Muriel returned with her father; and, after he had exchanged some friendly words with ‘Mein Herr’, and we had all been supplied with the needful ‘creature-comforts,’ the newcomer returned to the suggestive subject of Pocket-handkerchiefs.
‘You have heard of Fortunatus’s Purse, Miladi? Ah, so! Would you be surprised to hear that, with three of these leetle handkerchiefs, you shall make the Purse of Fortunatus, quite soon, quite easily?’
‘Shall I indeed?’ Lady Muriel eagerly replied, as she took a heap of them into her lap, and threaded her needle. ‘Please tell me how, Mein Herr! I’ll make one before I touch another drop of tea!’
‘You shall first,’ said Mein Herr, possessing himself of two of the handkerchiefs, spreading one upon the other, and holding them up by two corners, ‘you shall first join together these upper corners, the right to the right, the left to the left; and the opening between them shall be the mouth of the Purse.’
A very few stitches sufficed to carry out this direction. ‘Now, if I sew the other three edges together,’ she suggested, ‘the bag is complete?’
‘Not so, Miladi: the lower edges shall first be joined–ah, not so!’ (as she was beginning to sew them together). ‘Turn one of them over, and join the right lower corner of the one to the left lower corner of the other, and sew the lower edges together in what you would call the wrong way.’
‘I see!’ said Lady Muriel, as she deftly executed the order. ‘And a very twisted, uncomfortable, uncanny-looking bag it makes! But the moral is a lovely one. Unlimited wealth can only be attained by doing things in the wrong way! And how are we to join up these mysterious–no, I mean this mysterious opening?’ (twisting the thing round and round with a puzzled air.) ‘Yes, it is one opening. I thought it was two, at first.’
‘You have seen the puzzle of the Paper Ring?’ Mein Herr said, addressing the Earl. ‘Where you take a slip of paper, and join its ends together, first twisting one, so as to join the upper corner of one end to the lower corner of the other?‘
‘I saw one made, only yesterday,’ the Earl replied. ‘Muriel, my child, were you not making one, to amuse those children you had to tea?’
‘Yes, I know that Puzzle,’ said Lady Muriel. ‘The Ring has only one surface, and only one edge. It’s very mysterious!’
‘The bag is just like that, isn’t it?’ I suggested. ‘Is not the outer surface of one side of it continuous with the inner surface of the other side?’
‘So it is!’ she exclaimed. ‘Only it isn’t a bag, just yet. How shall we fill up this opening, Mein Herr?’
‘Thus!’ said the old man impressively, taking the bag from her, and rising to his feet in the excitement of the explanation. ‘The edge of the opening consists of four handkerchief-edges, and you can trace it continuously, round and round the opening: down the right edge of one handkerchief, up the left edge of the other, and then down the left edge of the one, and up the right edge of the other!’
‘So you can!’ Lady Muriel murmured thoughtfully, leaning her head on her hand, and earnestly watching the old man. ‘And that proves it to be only one opening!’
She looked so strangely like a child, puzzling over a difficult lesson, and Mein Herr had become, for the moment, so strangely like the old Professor, that I felt utterly bewildered: the ‘eerie’ feeling was on me in its full force, and I felt almost impelled to say ‘Do you understand it, Sylvie?’ However I checked myself by a great effort, and let the dream (if indeed it was a dream) go on to its end.
‘Now, this third handkerchief,’ Mein Herr proceeded, ‘has also four edges, which you can trace continuously round and round: all you need do is to join its four edges to the four edges of the opening. The Purse is then complete, and its outer surface–’
‘I see!’ Lady Muriel eagerly interrupted. ‘Its outer surface will be continuous with its inner surface! But it will take time. I’ll sew it up after tea.’ She laid aside the bag, and resumed her cup of tea. ‘But why do you call it Fortunatus’s Purse, Mein Herr?’
The dear old man beamed upon her, with a jolly smile, looking more exactly like the Professor than ever. ‘Don’t you see, my child–I should say Miladi? Whatever is inside that Purse, is outside it; and whatever is outside it, is inside it. So you have all the wealth of the world in that leetle Purse!’
Lewis Carroll demonstrates a problem with deciding an election by plurality of votes:
Four candidates are ranked by each of 11 electors, and each elector votes for his first choice. “Here A is considered best by three of the electors, and second by all the rest. It seems clear that he ought to be elected; and yet, by the above method, B would be the clear winner — a candidate who is considered worst by seven of the electors!”
“It is a matter for the deepest regret that Dodgson never completed the book he planned to write on this subject,” writes Michael Dummett. “Such was the lucidity of his exposition and mastery of this topic that it seems possible that, had he published it, the political history of Britain would have been significantly different.”
An astronomical oddity, from the Sidereal Messenger, June 1890:
On the evening of April 25th, 1889, at about 8:30 p.m., I was examining Saturn with a power of about 180 on a 4 1/8-inch achromatic by Brashear, when, much to my surprise, I found the shadow of the globe on the rings curved the wrong way, i.e. from the globe, as shown in the following drawing. Thinking my eyes might be deceiving me I called my wife, and without telling her what I had seen, requested her to describe the shape of the shadow. She described the shadow as having its right hand edge curved away from the planet.
I wrote to Professor Comstock of the Washburn Observatory about it, and was informed by him that while my observation of Saturn was unusual, it was far from being unprecedented; that the same appearance was observed in 1875 with the 26-inch achromatic at Washington, and that Webb, in ‘Celestial Objects for Common Telescopes,’ says: ‘The outline of this shadow has often been found curved the wrong way for its perspective.’ Professor Comstock also adds, ‘I do not know that any satisfactory explanation for this anomaly has ever been given.’
William Corliss notes a flurry of similar observations between 1886 and 1914. I think this must have been explained by now, but I haven’t been able to find a source.
(Jenks, Aldro; “On the Reversed Curvature of the Shadow on Saturn’s Rings,” Sidereal Messenger, 9:255, 1890.)
Awaiting the dawn sat three prisoners wary,
A trio of brigands named Tom, Dick and Mary.
Sunrise would signal the death knell of two;
Just one would survive, the question was who.
Young Mary sat thinking and finally spoke.
To the jailer she said, “You may think this a joke,
But it seems that my odds of surviving till tea
Are clearly enough just one out of three.
But one of my cohorts must certainly go,
Without question, that’s something I already know.
Telling the name of one who is lost
Can’t possibly help me. What could it cost?”
The shriveled old jailer himself was no dummy.
He thought, “But why not?” and pointed to Tommy.
“Now it’s just Dick and me!” Mary chortled with glee,
“One in two are my chances, and not one in three!”
Imagine the jailer’s chagrin, that old elf.
She’d tricked him. Or had she? Decide for yourself.
– Richard E. Bedient, “The Prisoner’s Paradox Revisited,” American Mathematical Monthly, March 1994
1234567891, 12345678901234567891, and 1234567891234567891234567891 are prime.
And so are
If the nth term of the Fibonacci series is prime, then n also is prime (where n > 4). For example, the 17th term, 1597, is prime, and 17 is prime.