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.”
See The Voting Paradox, The Referendum Paradox, and Stump Trouble.
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.
So are
19
197
1979
19793
197933
1979339
19793393 and
197933933.
And so are
742950290870000078092059247
742950290871010178092059247
742950290872020278092059247
742950290873030378092059247
742950290874040478092059247
742950290875050578092059247
742950290876060678092059247
742950290877070778092059247
742950290878080878092059247 and
742950290879090978092059247.
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.
05/27/2017 UPDATE: Further to the first series, 1979339333 is also prime! (Thanks, Alon.)
Given a book of matches, a box of thumbtacks, and a candle, how can you fix the candle to the wall so that its wax won’t drip onto the table below?
By 1941 Russian botanist Nikolai Vavilov had created the largest seed bank in the world, a collection of 400,000 seeds, roots, and fruits whose genetic material held the future of Soviet agriculture. Unfortunately it was located in Leningrad, which Hitler encircled that summer and began to starve.
The siege of Leningrad lasted two years and cost more than a million lives, and Vavilov’s scientists endured it surrounded by edible plants. “As they slowly starved, they refused to eat from any of their collection containers of rice, peas, corn and wheat,” two survivors remembered in 1993. “They chose torment and death in order to preserve Vavilov’s gene bank.”
The collection filled 16 rooms, in which no one was allowed to remain alone. Workers stored potatoes in the basement and guarded them in shifts, “numb with cold and emaciated from hunger.” Botanist Dmitri Ivanov died preserving thousands of packets of rice; peanut specialist Alexander Stchukin died at his writing table. In all, nine scientists and workers chose to die of starvation rather than eat the plants. Vavilov himself died in a labor camp in 1943, but today his bank is the largest collection of fruits and berries in the world.
(Thanks, Mike.)
When Richard Feynman won the Nobel Prize in 1965, CERN director Victor Weisskopf worried that he would be driven out of physics and into administration. He goaded Feynman into signing a wager before witnesses:
Mr. FEYNMAN will pay the sum of TEN DOLLARS to Mr. WEISSKOPF if at any time during the next TEN YEARS (i.e. before the THIRTY FIRST DAY OF DECEMBER of the YEAR ONE THOUSAND NINE HUNDRED AND SEVENTY FIVE), the said MR. FEYNMAN has held a ‘responsible position.’
The two agreed: “For the purpose of the aforementioned WAGER, the term ‘responsible position’ shall be taken to signify a position which, by reason of its nature, compels the holder to issue instructions to other persons to carry out certain acts, notwithstanding the fact that the holder has no understanding whatsoever of that which he is instructing the aforesaid persons to accomplish.”
Feynman, who once called administration an “occupational disease,” collected the $10 in 1976.
