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  #1  
Old 12-17-2005, 09:43 AM
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Thought you'd like this one, Kuan

Unwed Numbers

The mathematics of Sudoku, a puzzle that boasts "No math required!"
Brian Hayes

A few years ago, if you had noticed someone filling in a crossword puzzle with numbers instead of letters, you might well have looked askance. Today you would know that the puzzle is not a crossword but a Sudoku. The craze has circled the globe. It's in the newspaper, the bookstore, the supermarket checkout line; Web sites offer puzzles on demand; you can even play it on your cell phone.

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Just in case this column might fall into the hands of the last person in North America who hasn't seen a Sudoku, an example is given on the opposite page. The standard puzzle grid has 81 cells, organized into nine rows and nine columns and also marked off into nine three-by-three blocks. Some of the cells are already filled in with numbers called givens. The aim is to complete the grid in such a way that every row, every column and every block has exactly one instance of each number from 1 to 9. A well-formed puzzle has one and only one solution.

The instructions that accompany Sudoku often reassure the number-shy solver that "No mathematics is required." What this really means is that no arithmetic is required. You don't have to add up columns of figures; you don't even have to count. As a matter of fact, the symbols in the grid need not be numbers at all; letters or colors or fruits would do as well. In this sense it's true that solving the puzzle is not a test of skill in arithmetic. On the other hand, if we look into Sudoku a little more deeply, we may well find some mathematical ideas lurking in the background.

A Puzzling Provenance

The name "Sudoku" is Japanese, but the game itself is almost surely an American invention. The earliest known examples were published in 1979 in Dell Pencil Puzzles & Word Games, where they were given the title Number Place. The constructor of the puzzles is not identified in the magazine, but Will Shortz, the puzzles editor of The New York Times, thinks he has identified the author through a process of logical deduction reminiscent of what it takes to solve a Sudoku. Shortz examined the list of contributors in several Dell magazines; he found a single name that was always present if an issue included a Number Place puzzle, and never present otherwise. The putative inventor identified in this way was Howard Garns, an architect from Indianapolis who died in 1989. Mark Lagasse, senior executive editor of Dell Puzzle Magazines, concurs with Shortz's conclusion, although he says Dell has no records attesting to Garns's authorship, and none of the editors now on the staff were there in 1979.

The later history is easier to trace. Dell continued publishing the puzzles, and in 1984 the Japanese firm Nikoli began including puzzles of the same design in one of its magazines. (Puzzle publishers, it seems, are adept at the sincerest form of flattery.) Nikoli named the puzzle "suji wa dokushin ni kagiru," which I am told means "the numbers must be single"—single in the sense of unmarried. The name was soon shortened to Sudoku, which is usually translated as "single numbers." Nikoli secured a trademark on this term in Japan, and so later Japanese practitioners of sincere flattery have had to adopt other names. Ed Pegg, writing in the Mathematical Association of America's MAA Online, points out an ironic consequence: Many Japanese know the puzzle by its English name Number Place, whereas the English-speaking world prefers the Japanese term Sudoku.

The next stage in the puzzle's east-to-west circumnavigation was a brief detour to the south. Wayne Gould, a New Zealander who was a judge in Hong Kong before the British lease expired in 1997, discovered Sudoku on a trip to Japan and wrote a computer program to generate the puzzles. Eventually he persuaded The Times of London to print them; the first appeared in November 2004. The subsequent fad in the U.K. was swift and intense. Other newspapers joined in, with The Daily Telegraph running the puzzle on its front page. There was boasting about who had the most and the best Sudoku, and bickering over the supposed virtues of handmade versus computer-generated puzzles. In July 2005 a Sudoku tournament was televised in Britain; the event was promoted by carving a 275-foot grid into a grassy hillside near Bristol. (It soon emerged that this "world's largest Sudoku" was defective.)

Sudoku came back to the U.S. in the spring of 2005. Here too the puzzle has become a popular pastime, although perhaps not quite the all-consuming obsession it was in the U.K. I don't believe anyone will notice a dip in the U.S. gross domestic product as a result of this mass distraction. On the other hand, I must report that my own motive for writing on the subject is partly to justify the appalling number of hours I have squandered solving Sudoku.

(more at http://www.americanscientist.org/template/AssetDetail/assetid/48550?&print=yes)
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Old 12-17-2005, 10:14 AM
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Hah yes, there's a Sudoku puzzle right next to the crossword in our local paper. Monday-Wednesday you can solve in about 5, but then it gets harder. It's a curious little game of deduction and to be honest, the amount of time I spend doing puzzles these days is limited to the length of the kid's attention span, and of course the time he takes to finish his pancakes at McFood.

Wanna real challenge? Try magic squares!
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  #3  
Old 12-28-2005, 05:26 PM
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Here's another, from Wikipedia.

The Doomsday argument
This article introduces the Doomsday argument (DA) in four alternative ways:

For a description of the DA without mathematics see the analogy-to-cricket section.
For a very simplified numerical example, partially based on Korb’s refutation[1] see the two-case section.
For a general overview, with numerical examples see the next section.
For Gott's mathematical development of the Bayesian argument (using simple calculus) see the Vague Prior section.
[edit]
Numerical estimate of Doomsday
Let us imagine our fractional position f = n/N along the chronological list of all the humans who will ever be born, where:

n is our absolute position from the beginning of the list.
N is the total number of humans.
The Copernican principle suggests that we are equally likely (along with the other N-1 humans) to find ourselves at any position n, so our fractional position f is uniformly distributed on the interval (0,1] prior to learning our absolute position.

Let us further assume that our fractional position f is uniformly distributed on (0,1] even after we learn of our absolute position n. This is equivalent to the assumption that we have no prior information about the total number of humans, N.

Now, we can say with 95% confidence that f = n/N is within the interval (0.05,1]. In other words we are 95% certain that we are within the last 95% of all the humans ever to be born. Given our absolute position n, this implies an upper bound for N obtained by rearranging

n / N > 0.05
to give

N < 20n.
If we assume that 60 billion humans have been born so far (Leslie's figure) then we can say with 95% confidence that the total number of humans, N, will be less than 20·60 = 1200 billion.

Assuming that the world population stabilizes at 10 billion and a life expectancy of 80 years, one can calculate how long it will take for the remaining 1140 billion humans to be born. The argument predicts, with 95% confidence, that humanity will disappear within 9120 years. Depending on your projection of world population in the forthcoming centuries, your estimates might vary, but the main point of the argument is that we are likely to disappear rather soon.

[edit]
Remarks

The Wheel of fortune, 1510: Contrasted with Sapientia (wisdom) the blindfolded Goddess Fortuna and the Tarot wheel symbolize the Medieval belief in impermanence, representing the principle: This too shall pass.The step that converts N into an extinction time depends upon a finite human lifespan. If immortality becomes common, and the birth rate drops to zero, N will never be reached1.
The total number of humans born so far may depend on one's definition of "human".
By counting the number of human consciousnesses as states, the Doomsday argument (DA) imputes a special value to the human mind that rejects the Copernican tradition's mediocrity principle.
A precise formulation of the DA requires the Bayesian interpretation of probability, which is widely, if not universally, accepted.
Even among Bayesians some of the assumptions of the argument's logic would not be acceptable; for instance, the fact that it is applied to a temporal phenomenon (how long something lasts) means that N's distribution simultaneously represents an "aleatory probability" (as a future event), and an "epistemic probability" (as a decided value about which we are uncertain).
The U(0,1] f distribution is derived from two choices, which whilst being the default are also arbitrary:
The principle of indifference, so that it is as likely for any other randomly selected person to be born after you as before you.
The assumption of no 'prior' knowledge on the distribution of N.
[edit]
Simplification: two possible total number of humans
Assume for simplicity that the total number of humans who will ever be born is 60 billion (N1), or 6,000 billion (N2). If there is no prior knowledge of the position that a currently living individual, X, has in the history of humanity, we may instead compute how many humans were born before X, and arrive at (say) 59,854,795,447, which would roughly place X amongst the first 60 billion humans who have ever lived.

Now, if we assume that the number of humans who will ever be born equals N1, the probability that X is amongst the first 60 billion humans who have ever lived is of course 100%. However, if the number of humans who will ever be born equals N2, then the probability that X is amongst the first 60 billon humans who have ever lived is only 1%, such that the total number of humans who will ever be born is more likely to be much more closer to 60 billion than to 6,000 billion. In essence the DA therefore suggests that human extinction is more likely to occur sooner rather than later.

It is possible to sum the probabilities for each value of N and therefore to compute a statistical 'confidence limit' on N. For example, taking the numbers above, it is 99% certain that N is smaller that 6,000 billion.

[edit]
What the argument is not
The Doomsday argument (DA) does not say that humanity cannot or will not exist indefinitely. It does not put any upper limit on the number of humans that will ever exist, nor provide a date for when humanity will become extinct.

An abbreviated form of the argument does make these claims, by confusing probability with certainty. However, the actual DA's conclusion is:

There is a 95% chance of extinction within 9120 years.
The DA gives a 5% chance that humans will still be thriving circa 11125 AD. (These dates are based on the assumptions above; the precise numbers vary among specific Doomsday arguments.)
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Old 12-28-2005, 05:31 PM
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i read a book on time travel and in it was an eqaution to find out how long you would live . and part of it was if you've made it past 37 you will be good

ill find the equation in the book i borrowed from the library .. pretty neat stuff
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Old 08-26-2006, 11:13 AM
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Another Kuan moment

MANIFOLD DESTINY
by SYLVIA NASAR AND DAVID GRUBER
A legendary problem and the battle over who solved it.
Issue of 2006-08-28
Posted 2006-08-21

On the evening of June 20th, several hundred physicists, including a Nobel laureate, assembled in an auditorium at the Friendship Hotel in Beijin for a lecture by the Chinese mathematician Shing-Tung Yau. In the late nineteen-seventies, when Yau was in his twenties, he had made a series o breakthroughs that helped launch the string-theory revolution in physics and earned him, in addition to a Fields Medal—the most coveted award i mathematics—a reputation in both disciplines as a thinker of unrivalled technical power
Yau had since become a professor of mathematics at Harvard and the director of mathematics institutes in Beijing and Hong Kong, dividing his time between the United States and China. His lecture at the Friendship Hotel was part of an international conference on string theory, which he had organized with the support of the Chinese government, in part to promote the country’s recent advances in theoretical physics. (More than six thousand students attended the keynote address, which was delivered by Yau’s close friend Stephen Hawking, in the Great Hall of the People.) The subject of Yau’s talk was something that few in his audience knew much about: the Poincaré conjecture, a century-old conundrum about the characteristics of three-dimensional spheres, which, because it has important implications for mathematics and cosmology and because it has eluded all attempts at solution, is regarded by mathematicians as a holy grail.
Yau, a stocky man of fifty-seven, stood at a lectern in shirtsleeves and black-rimmed glasses and, with his hands in his pockets, described how two of his students, Xi-Ping Zhu and Huai-Dong Cao, had completed a proof of the Poincaré conjecture a few weeks earlier. “I’m very positive about Zhu and Cao’s work,” Yau said. “Chinese mathematicians should have every reason to be proud of such a big success in completely solving the puzzle.” He said that Zhu and Cao were indebted to his longtime American collaborator Richard Hamilton, who deserved most of the credit for solving the Poincaré. He also mentioned Grigory Perelman, a Russian mathematician who, he acknowledged, had made an important contribution. Nevertheless, Yau said, “in Perelman’s work, spectacular as it is, many key ideas of the proofs are sketched or outlined, and complete details are often missing.” He added, “We would like to get Perelman to make comments. But Perelman resides in St. Petersburg and refuses to communicate with other people.”
For ninety minutes, Yau discussed some of the technical details of his students’ proof. When he was finished, no one asked any questions. That night, however, a Brazilian physicist posted a report of the lecture on his blog. “Looks like China soon will take the lead also in mathematics,” he wrote.

Grigory Perelman is indeed reclusive. He left his job as a researcher at the Steklov Institute of Mathematics, in St. Petersburg, last December; he ha few friends; and he lives with his mother in an apartment on the outskirts of the city. Although he had never granted an interview before, he wa cordial and frank when we visited him, in late June, shortly after Yau’s conference in Beijing, taking us on a long walking tour of the city. “I’ looking for some friends, and they don’t have to be mathematicians,” he said. The week before the conference, Perelman had spent hours discussin the Poincaré conjecture with Sir John M. Ball, the fifty-eight-year-old president of the International Mathematical Union, the discipline’s influentia professional association. The meeting, which took place at a conference center in a stately mansion overlooking the Neva River, was highly unusual At the end of May, a committee of nine prominent mathematicians had voted to award Perelman a Fields Medal for his work on the Poincaré, and Bal had gone to St. Petersburg to persuade him to accept the prize in a public ceremony at the I.M.U.’s quadrennial congress, in Madrid, on August 22nd
The Fields Medal, like the Nobel Prize, grew, in part, out of a desire to elevate science above national animosities. German mathematicians were excluded from the first I.M.U. congress, in 1924, and, though the ban was lifted before the next one, the trauma it caused led, in 1936, to the establishment of the Fields, a prize intended to be “as purely international and impersonal as possible.”
However, the Fields Medal, which is awarded every four years, to between two and four mathematicians, is supposed not only to reward past achievements but also to stimulate future research; for this reason, it is given only to mathematicians aged forty and younger. In recent decades, as the number of professional mathematicians has grown, the Fields Medal has become increasingly prestigious. Only forty-four medals have been awarded in nearly seventy years—including three for work closely related to the Poincaré conjecture—and no mathematician has ever refused the prize. Nevertheless, Perelman told Ball that he had no intention of accepting it. “I refuse,” he said simply.
Over a period of eight months, beginning in November, 2002, Perelman posted a proof of the Poincaré on the Internet in three installments. Like a sonnet or an aria, a mathematical proof has a distinct form and set of conventions. It begins with axioms, or accepted truths, and employs a series of logical statements to arrive at a conclusion. If the logic is deemed to be watertight, then the result is a theorem. Unlike proof in law or science, which is based on evidence and therefore subject to qualification and revision, a proof of a theorem is definitive. Judgments about the accuracy of a proof are mediated by peer-reviewed journals; to insure fairness, reviewers are supposed to be carefully chosen by journal editors, and the identity of a scholar whose pa-per is under consideration is kept secret. Publication implies that a proof is complete, correct, and original.
By these standards, Perelman’s proof was unorthodox. It was astonishingly brief for such an ambitious piece of work; logic sequences that could have been elaborated over many pages were often severely compressed. Moreover, the proof made no direct mention of the Poincaré and included many elegant results that were irrelevant to the central argument. But, four years later, at least two teams of experts had vetted the proof and had found no significant gaps or errors in it. A consensus was emerging in the math community: Perelman had solved the Poincaré. Even so, the proof’s complexity—and Perelman’s use of shorthand in making some of his most important claims—made it vulnerable to challenge. Few mathematicians had the expertise necessary to evaluate and defend it.
After giving a series of lectures on the proof in the United States in 2003, Perelman returned to St. Petersburg. Since then, although he had continued to answer queries about it by e-mail, he had had minimal contact with colleagues and, for reasons no one understood, had not tried to publish it. Still, there was little doubt that Perelman, who turned forty on June 13th, deserved a Fields Medal. As Ball planned the I.M.U.’s 2006 congress, he began to conceive of it as a historic event. More than three thousand mathematicians would be attending, and King Juan Carlos of Spain had agreed to preside over the awards ceremony. The I.M.U.’s newsletter predicted that the congress would be remembered as “the occasion when this conjecture became a theorem.” Ball, determined to make sure that Perelman would be there, decided to go to St. Petersburg.
Ball wanted to keep his visit a secret—the names of Fields Medal recipients are announced officially at the awards ceremony—and the conference center where he met with Perelman was deserted. For ten hours over two days, he tried to persuade Perelman to agree to accept the prize.

more at the New Yorker
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Old 08-26-2006, 12:32 PM
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Old 08-27-2006, 04:56 PM
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Topology is way interesting but way over my head. There are millions of dollars in prizes out there for solving math problems. One of the ones which interests me has to do with NP completeness.

I guess the work of Yau's students dispels myths about how Chinese students don't learn how to be creative.
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Old 08-27-2006, 07:56 PM
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Quote:
Originally Posted by Kuan View Post
Topology is way interesting but way over my head. There are millions of dollars in prizes out there for solving math problems. One of the ones which interests me has to do with NP completeness.

I guess the work of Yau's students dispels myths about how Chinese students don't learn how to be creative.
I'll bet some Roman somewhere said, "Those Germanic tribes, all they know about chariot-making is copying."

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