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Rubik's God Number = 20

The Rubik cube has 43,252,003,274,489,856,000 positions. What is the minimum number of moves that will solve the puzzle, no matter which configuration you start from? The number is jokingly called the 'God Number' because an all-powerful deity would know what it is. Until recently, the best bounds were at least 18 and no more than 26. Now a team haded by Morley Davidson of Kent State University has proved that the long-sought answer is 20 precisely.

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Mathematics News Worldwide

€8000 in Prizes for Magic Squares

Magic squares are a well known topic in recreational mathematics, but there are many unsolved and difficult mathematical problems in this area. Christian Boyer has announced twelve prizes totaling €8,000 and 12 bottles of champagne for the solutions to twelve of these, on his Multimagic Squares website. For instance, does there exist a 3x3 magic square, all of whose entries are perfect squares? A 5x5 bimagic square using distinct integers?

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Mathematics News Worldwide

First Million-dollar Clay Prize Awarded

Grigoriy Perelman has now (18 March 2010) been officially awarded the first Clay Mathematics Institute Millennium Prize, worth $1,000,000, for his proof of the Poincaré Conjecture. This long-sought theorem characterises the 3-sphere as the only connected compact 3-manifold without boundary in which every closed curve can be deformed continuously to a point. Perelman previously declined a Fields Medal for the same achievement, preferring to remain out of the public eye.

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Mathematics News Worldwide

Crocheting the Hyperbolic Plane

wins prize for oddest book title

The book Crocheting Adventures with Hyperbolic Planes, by Daina Taimina, has won the Diamond Prize for oddest book title of the year.

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Mathematics News Worldwide

How Hot is Your Pepper?

A new mathematical model makes it possible to determine how hot a chili pepper is. The sensation of 'heat' is caused by capsaicinoid compounds in the pepper, which bind to taste receptors in the mouth and throat. Previous ways to find out how hot a pepper is involved detailed measurements of its capsaicinoids. The new method, invented by Kenneth Busch at Baylor University, fits data from known capsaicinoids and is simpler and quicker.

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Mathematics News Worldwide

Golden Ratio and E8 in Quantum Mechanics

The golden ratio (1.618034...) is one of the most intriguing numbers in mathematics, and entire books have been written about it. Many myths surround this number, such as its role in aesthetics, which is generally exaggerated. However, its mathematical importance is undeniable.

A mathematical prediction that this ratio should occur in a particular quantum-mechanical system has now been verified by Radu Coldea and colleagues at Oxford University, in cobalt niobate. By applying a magnetic field, they made the state of a magnetic chain in the crystal become 'quantum critical'. The two lowest frequencies of vibration of the chain were predicted to be in the golden ratio, and the experiments confirm this. The detailed theory involves the exceptional Lie group E8, a strange 248-dimensional group of symmetries that keeps turning up in quantum physics.

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Mathematics News Worldwide

Consider a Spherical Cow...

So goes the old joke about the farmer who hired a mathematician to advise him on how to increase milk production. Now University of Vermont mathematician Peter Dodds has given the spherical cow a new lease of life. The question is: how does an animal's size vary with its metabolic rate? The classic answer is a 3/4 power law, obtained in 1932 by the Swiss chemist Max Kleiber. He plotted the body weight of 13 mammals against their resting metabolism on a logarithmic scale. The result was a line with slope 0.73. He rounded this up to 0.75, and the famous 3/4 power law came into being.

The exponent 3/4 has long puzzled biologists (and mathematicians) because the obvious value is 2/3. To see why, consider a spherical cow. The surface area of this rotund beast varies as the square of its radius, but the volume increases as the cube of the radius. The metabolic rate should be proportional to the surface area, because heat is lost through the surface; the mass should be proportional to the volume. Putting this together, we get a 2/3 power law.

Many explanations have been invoked for the 3/4 power law rather than the 2/3 'spherical cow' version. The most recent invokes a fractal cow instead, considering the network of blood vessels, but this approach has proved controversial.

Dodds argues that Kleiber's data was poor and the 3/4 power law is a myth: the data fit a 2/3 power law just as well. By reworking the fractal approach, but making different assumptions, he obtains a 2/3 power law. So the fractal cow agrees with the spherical cow.

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Warwick News

The Trillion Triangle Problem

BILL HART

Bill Hart of the Warwick Mathematics Institute is part of a team of mathematicians from North America, Europe, Australia, and South America who have found the congruent numbers up to one trillion. This is a significant contribution to a thousand-year-old mathematical problem posed by al-Karaji: find the integers n for which there is a rational square a2 such that a2+n and a2-n are also rational squares.

Although the equivalence is not obvious, congruent numbers can also be defined as those integers which are areas of right-angled triangles with rational sides.

For example: (41/12)2+5=(49/12)2, (41/12)2-5=(31/12)2 so 5 is a congruent number.The corresponding triangle has sides 20/3, 3/2, and 41/6. The familiar 3-4-5 triangle has area 1/2 x 3 x 4 = 6, so 6 is a congruent number.

Among the smaller integers, 1, 2, 3, 4 are not congruent, but 5, 6, and 7 are. It is not straightforward to decide whether a number is congruent—for instance, 157 is congruent, but the simplest right-angled triangle with area 157 has hypotenuse

2244035177043369699245755130906674863160948472041

divided by

8912332268928859588025535178967163570016480830

The best test currently known depends on an unproved conjecture, the Birch—Swinnerton-Dyer Conjecture, which is one of the Clay Millennium Mathematics Prize problems, with a million dollars offered for a proof or disproof.

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Mathematics News Worldwide

What's Purple and Commutes?

Pablo Flores Martínez at the University of Granada has collected 4000 mathematical jokes and cartoons over the past 14 years, an he uses them in his course 'Didactics of Mathematics'.

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Mathematics News Worldwide

Supercomputers and the Big Bang

A new mathematical model is helping to untangle some of the scientific riddles associated with the Big Bang theory of the origin of the universe. This model incorporates many physical factors, such as gas motion, radiation transport, chemical kinetics, star clustering, and dark matter. The main new feature is that all of these processes are tightly coupled in the model. Daniel R. Reynolds at SMU has collaborated with astrophysicists at the University of California at San Diego as part of an NSF project to simulate cosmic reionization, believed to have occurred from 380000 years to 400 million years after the birth of the universe.

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Mathematics News Worldwide

123 Billion More Digits of Pi

Computer scientist Fabrice Bellard has calculated pi to 2.7 trillion decimal digits, 123 billion more than the previous record. Unlike most recent calculations of the digits of pi, his method did not use a supercomputer: instead, it ran on a standard desktop machine in Linux. It took 131 days, whereas the previous record (obtained by Daisuke Takahashi at the University of Tsukuba) took 29 hours.

Although mathematicians do not consider the digits of pi to have any special significance, beyond a conjecture that they satisfy all the standard statistical tests for being random, calculations of this kind are an effective way to test computers, and the algorithms used often have independent interest. It's how to get the result that matters, rather than the result itself.

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