Tuesday, 31 January 2012
Monday, 30 January 2012
Sunday, 29 January 2012
Saturday, 28 January 2012
Friday, 27 January 2012
Thursday, 26 January 2012
Wednesday, 25 January 2012
Tuesday, 24 January 2012
Flipping Pennies
The Jan 1, 2012 issue of The College Mathematics Journal featured articles inspired by the late-great Martin Gardner. One by Ian Stewart begins with the Three Penny Puzzle invented by Gardner and Karl Fulves:
Gardner’s three-penny trick:Not being very good at card tricks and such, I would sometimes perform this trick in study hall or odd class moments to amuse my students, and then challenge them to duplicate the trick. I avoided doing it more than once in order not to have them catch on to the fact that the same pattern always works (why children). In all the many times I did it, and sometimes had students discover the logic, I don't think I ever asked them if they could do the same with four pennies, or five, or n. That is, for n Pennies, what is the minimal number of blind-flips f(n) to get them all alike, and what is the sequence of flips.
The trick is performed by a blindfolded magician. A volunteer places three pennies
in a row, and chooses at will whether each coin shows heads or tails. However, both heads and tails must appear, otherwise the trick ends before it begins. The magician announces that even though she cannot see the coins, she will give instructions to turn coins over so that all three coins show the same face, heads or tails.
The instructions are:
1. Flip the left-hand coin.
2. Flip the middle coin.
3. Flip the left-hand coin.
After steps 1 and 2 the magician asks whether all three coins show the same face,
and if the answer is ‘yes’, the trick stops, otherwise the magician requests the third flip.
Although it is plausible that enough flips will eventually
get all coins the same way up, it is a little surprising that at most three flips are needed.
I assume that the questions from the solver can only be "are all coins alike now?"
Monday, 23 January 2012
Sunday, 22 January 2012
Saturday, 21 January 2012
Friday, 20 January 2012
Thursday, 19 January 2012
Wednesday, 18 January 2012
Tuesday, 17 January 2012
Monday, 16 January 2012
Sunday, 15 January 2012
Saturday, 14 January 2012
Friday, 13 January 2012
Thursday, 12 January 2012
Just came across a James Tanton blog that has some nice extensions of the old joke about bad cancellations in fractions. I used to show my students that you could simplify 16/64 by just crossing out the sixes to get 1/4. Among the similar type examples that he gives in the blog is the rather clever:
Cancel all the common numerals top and bottom and get...
See it all here, and enjoy
Cancel all the common numerals top and bottom and get...
See it all here, and enjoy
I came across an account of this interview in "Eurekas and Euphorias" by Walter Gratzer, but found this copy on a web page from Rutgers which has not been updated since 2004.... Dirac was famous as a man of extremely few words:
I been hearing about a fellow they have up at the U. this spring --- a mathematical physicist, or something, they call him --- who is pushing Sir Isaac Newton, Einstein and all the others off the front page. So I thought I better go up and interview him for the benefit of State Journal readers, same as I do all other top notchers. His name is Dirac and he is an Englishman. He has been giving lectures for the intelligentsia of math and physics departments --- and a few other guys who got in by mistake.
So the other afternoon I knocks at the door of Dr. Dirac's office in Sterling Hall and a pleasant voice says "Come in." And I want to say here and now that this sentence "come in" was about the longest one emitted by the doctor during our interview. He sure is all for efficiency in conversation. It suits me. I hate a talkative guy. I found the doctor a tall youngish-looking man, and the minute I seen the twinkle in his eye I knew I was going to like him. His friends at the U. say he is a real fellow too and a good company on a hike --- if you can keep him in sight, that is.
The thing that hit me in the eye about him was that he did not seem to be at all busy. Why if I went to interview an American scientist of his class --- supposing I could find one --- I would have to stick around an hour first. Then he would blow in carrying a big briefcase, and while he talked he would be pulling lecture notes, proof, reprints, books, manuscript, or what have you out of his bag. But Dirac is different. He seems to have all the time there is in the world and his heaviest work is looking out the window. If he is a typical Englishman it's me for England on my next vacation!
Then we sat down and the interview began.
"Professor," says I, "I notice you have quite a few letters in front of your last name. Do they stand for anything in particular?"
"No," says he.
"You mean I can write my own ticket?"
"Yes," says he.
"Will it be all right if I say that P.A.M. stands for Poincare' Aloysius Mussolini?"
"Yes," says he.
"Fine," says I, "We are getting along great! Now doctor will you give me in a few words the low-down on all your investigations?"
"No," says he.
"Good," says I. "Will it be all right if I put it this way --- `Professor Dirac solves all the problems of mathematical physics, but is unable to find a better way of figuring out Babe Ruth's batting average'?"
"Yes," says he.
"What do you like best in America?", says I.
"Potatoes," says he.
"Same here," says I. "What is your favorite sport?"
"Chinese chess," says he.
That knocked me cold! It was sure a new one on me! Then I went on: "Do you go to the movies?"
"Yes," says he.
"When?", says I.
"In 1920 --- perhaps also in 1930," says he.
"Do you like to read the Sunday comics?"
"Yes," says he, warming up a bit more than usual.
"This is the most important thing yet, doctor," says I. "It shows that me and you are more alike than I thought. And now I want to ask you something more: They tell me that you and Einstein are the only two real sure-enough high-brows and the only ones who can really understand each other. I wont ask you if this is straight stuff for I know you are too modest to admit it. But I want to know this --- Do you ever run across a fellow that even you can't understand?"
"Yes," says he.
"This well make a great reading for the boys down at the office," says I. "Do you mind releasing to me who he is?"
"Weyl," says he.
The interview came to a sudden end just then, for the doctor pulled out his watch and I dodged and jumped for the door. But he let loose a smile as we parted and I knew that all the time he had been talking to me he was solving some problem that no one else could touch.
But if that fellow Professor Weyl ever lectures in this town again I sure am going to take a try at understanding him! A fellow ought to test his intelligence once in a while.
ROUNDY INTERVIEWS PROFESSOR DIRAC
An Enjoyable Time Is Had By All
By RoundySo the other afternoon I knocks at the door of Dr. Dirac's office in Sterling Hall and a pleasant voice says "Come in." And I want to say here and now that this sentence "come in" was about the longest one emitted by the doctor during our interview. He sure is all for efficiency in conversation. It suits me. I hate a talkative guy. I found the doctor a tall youngish-looking man, and the minute I seen the twinkle in his eye I knew I was going to like him. His friends at the U. say he is a real fellow too and a good company on a hike --- if you can keep him in sight, that is.
The thing that hit me in the eye about him was that he did not seem to be at all busy. Why if I went to interview an American scientist of his class --- supposing I could find one --- I would have to stick around an hour first. Then he would blow in carrying a big briefcase, and while he talked he would be pulling lecture notes, proof, reprints, books, manuscript, or what have you out of his bag. But Dirac is different. He seems to have all the time there is in the world and his heaviest work is looking out the window. If he is a typical Englishman it's me for England on my next vacation!
Then we sat down and the interview began.
"Professor," says I, "I notice you have quite a few letters in front of your last name. Do they stand for anything in particular?"
"No," says he.
"You mean I can write my own ticket?"
"Yes," says he.
"Will it be all right if I say that P.A.M. stands for Poincare' Aloysius Mussolini?"
"Yes," says he.
"Fine," says I, "We are getting along great! Now doctor will you give me in a few words the low-down on all your investigations?"
"No," says he.
"Good," says I. "Will it be all right if I put it this way --- `Professor Dirac solves all the problems of mathematical physics, but is unable to find a better way of figuring out Babe Ruth's batting average'?"
"Yes," says he.
"What do you like best in America?", says I.
"Potatoes," says he.
"Same here," says I. "What is your favorite sport?"
"Chinese chess," says he.
That knocked me cold! It was sure a new one on me! Then I went on: "Do you go to the movies?"
"Yes," says he.
"When?", says I.
"In 1920 --- perhaps also in 1930," says he.
"Do you like to read the Sunday comics?"
"Yes," says he, warming up a bit more than usual.
"This is the most important thing yet, doctor," says I. "It shows that me and you are more alike than I thought. And now I want to ask you something more: They tell me that you and Einstein are the only two real sure-enough high-brows and the only ones who can really understand each other. I wont ask you if this is straight stuff for I know you are too modest to admit it. But I want to know this --- Do you ever run across a fellow that even you can't understand?"
"Yes," says he.
"This well make a great reading for the boys down at the office," says I. "Do you mind releasing to me who he is?"
"Weyl," says he.
The interview came to a sudden end just then, for the doctor pulled out his watch and I dodged and jumped for the door. But he let loose a smile as we parted and I knew that all the time he had been talking to me he was solving some problem that no one else could touch.
But if that fellow Professor Weyl ever lectures in this town again I sure am going to take a try at understanding him! A fellow ought to test his intelligence once in a while.
Wednesday, 11 January 2012
Tuesday, 10 January 2012
Monday, 9 January 2012
Sunday, 8 January 2012
Saturday, 7 January 2012
Friday, 6 January 2012
Thursday, 5 January 2012
Wednesday, 4 January 2012
Tuesday, 3 January 2012
Monday, 2 January 2012
Sunday, 1 January 2012
2012 is NOT Weird...
and you are not likely to ever see a year number that is weird in your lifetime. Now that is weird.
The early meaning of weird was related to turning, or falling, from the Ind-European root wer, which makes its way into lots of mathematical words; converse, inverse, diverge, etc. That meaning seems to have quickly given way to things "befalling" an individual, and were tied to fate. Then Shakespeare, the Matt Groening of the late 16th century, decided to use the word to describe one of his three witches, and as everybody knows, it caused another "turn" in the common usage of the word and it came to mean strange or unusual.
So finally I can get away from literature, about which I know very little, and talk about number theory, about which I know...... Oh never mind.
So anyway, while some folks think ALL numbers are a little weird, in number theory the term is usually applied to a subset of the abundant numbers. Ok, a little background....
Perfect numbers are integers that are the sum of their proper divisors. Six is the smallest since 6 = 1+2+3. The next few, all known to the Greeks are 28, 496, and 8128. There is a connection, known to Euclid, that ties certain prime numbers to the perfect numbers.
OK, so not too many numbers are perfect. What about the rest? Well it's one of those Goldilocks things. (College profs prefer to talk about trichotomy laws, but I'm a fan of nursery rhymes)
If the sum is too small, smaller than the integer itself, the number is called deficient. If the sum is too large, the number is called abundant. But if the total is "Just Right" then baby bear eats it all up because it is perfect.
So what about weird numbers? Well, if you take your typical abundant number, 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102 ... and more here , and look at the divisors you can almost always find a subset of the proper divisors that does add up to the original integer.
The divisors of 12, for example, are 1,2,3,4,6 and you can use 6+4+2 to get 12. You can do that for most all of them. But once in a while you come across a strange and unusual number (dare I say weird?).. the smallest one is 70, in which there is no subset of the divisors that will add up to the original number. Seventy's divisors are 1, 2, 5, 7, 10, 14, and 35 go ahead, try your luck. I'll wait...Dum de dum... tra la...tra la...... Dum de dum de dum...
Ok, Give up yet?
So that's what a weird number is, and there really aren't many of them... I mean relatively. Actually there may be an infinite number of them, we don't know. How unusual? Well, between the year 70 and the year 4036,(let's see, how old will I be then...oh never mind) there is only one more weird number (and as I told you above, it's not 2012). If you want to look for it, look to the past, not the future. And don't even bother with the odds. We haven't found any odd ones yet, but then we haven't proven there are not any yet either, so if you want to tackle that problem, just look farther out in the bigger numbers...bigger than 232 at least.
The last three years have included a prime number (always deficient) , 2011, and an abundant composite ,2010, as well as a deficient composite , 2012.
The early meaning of weird was related to turning, or falling, from the Ind-European root wer, which makes its way into lots of mathematical words; converse, inverse, diverge, etc. That meaning seems to have quickly given way to things "befalling" an individual, and were tied to fate. Then Shakespeare, the Matt Groening of the late 16th century, decided to use the word to describe one of his three witches, and as everybody knows, it caused another "turn" in the common usage of the word and it came to mean strange or unusual.
So finally I can get away from literature, about which I know very little, and talk about number theory, about which I know...... Oh never mind.
So anyway, while some folks think ALL numbers are a little weird, in number theory the term is usually applied to a subset of the abundant numbers. Ok, a little background....
Perfect numbers are integers that are the sum of their proper divisors. Six is the smallest since 6 = 1+2+3. The next few, all known to the Greeks are 28, 496, and 8128. There is a connection, known to Euclid, that ties certain prime numbers to the perfect numbers.
OK, so not too many numbers are perfect. What about the rest? Well it's one of those Goldilocks things. (College profs prefer to talk about trichotomy laws, but I'm a fan of nursery rhymes)
If the sum is too small, smaller than the integer itself, the number is called deficient. If the sum is too large, the number is called abundant. But if the total is "Just Right" then baby bear eats it all up because it is perfect.
So what about weird numbers? Well, if you take your typical abundant number, 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102 ... and more here , and look at the divisors you can almost always find a subset of the proper divisors that does add up to the original integer.
The divisors of 12, for example, are 1,2,3,4,6 and you can use 6+4+2 to get 12. You can do that for most all of them. But once in a while you come across a strange and unusual number (dare I say weird?).. the smallest one is 70, in which there is no subset of the divisors that will add up to the original number. Seventy's divisors are 1, 2, 5, 7, 10, 14, and 35 go ahead, try your luck. I'll wait...Dum de dum... tra la...tra la...... Dum de dum de dum...
Ok, Give up yet?
So that's what a weird number is, and there really aren't many of them... I mean relatively. Actually there may be an infinite number of them, we don't know. How unusual? Well, between the year 70 and the year 4036,(let's see, how old will I be then...oh never mind) there is only one more weird number (and as I told you above, it's not 2012). If you want to look for it, look to the past, not the future. And don't even bother with the odds. We haven't found any odd ones yet, but then we haven't proven there are not any yet either, so if you want to tackle that problem, just look farther out in the bigger numbers...bigger than 232 at least.
The last three years have included a prime number (always deficient) , 2011, and an abundant composite ,2010, as well as a deficient composite , 2012.
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