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Weekdays 15:45-16:00
How many shuffles does it take to randomise a pack of cards? How many colours do cartographers need? What is the largest prime number? A quirky look at five numbers that lie at the heart of some of the trickiest problems in mathematics.
LISTEN AGAINListen 15 min
1. The Number Four
2. The Number Seven
3. The Largest Prime
4. Kepler's Conjecture
5. Game Theory
Simon Singh
Monday to Friday 27 - 31 October 2003
Another 5 Numbers
Following on from the original series, exploring numbers from zero to infinity, Simon Singh uncovers the mathematical, social and scientific history and significance of another five numbers, over five exploratory episodes.

These numbers have been at the centre of some of mathematics' most challenging problems and Simon reveals their historical, practical and esoteric merits. Along the way we get to grips with the search for the largest prime number, which serves as a tool for the encoding and encryption of information on the internet. Whilst in the casinos of America, mathematicians are making gambling heads roll: why is the number 7 so influential in properly shuffling a pack of cards?

Programme 1: The Number Four

The journey begins with the number 4, which for over a century has fuelled one of maths' most elusive problems - so easy to state but nearly impossible to prove: is it true that any map can be coloured with just 4 colours so that no two neighbouring countries have the same colour? The problem has tested some of the most imaginative minds - from problem solvers such as Lewis Carroll to modern mathematicians - and the eventual solution has aided the design of some of the world's most complex air and road networks.
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Programme 2: The Number Seven
In 1990, American mathematicians Persi Diaconis and David Bayer suggested that 7 shuffles are sufficient to achieve an acceptable degree of randomness in a deck of 52 cards. With 6 or fewer shuffles, the original order of the cards is still strongly in evidence and beyond 7, nothing is gained in terms of increased randomness.
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Programme 3: The Largest Prime

Think of a number. Any number. Chances are you haven't plumped for 213,466,917 -1. To get this, you would need to keep multiplying 2 by itself 13,466,917 times, and then subtract 1 from the result. When written down it's 4,053,900 digits long and fills 2 telephone directories. So, as you can imagine, it's not the kind of number you're likely to stumble over often. Unless you're Bill Gates checking your bank statement at the end of the month.
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Programme 4:
Kepler's Conjecture

Johannes Kepler experimented with different ways of stacking spheres. He concluded that the "face-centred cubic lattice" was best. Using this method, Kepler calculated that the packing efficiency rose to 74%, constituting the highest efficiency you could ever get. But, how to prove it?
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Programme 5: Game Theory

In 2000, the UK government received a windfall of around £23 billion from its auction of third generation (3G) mobile phone licences. This astronomical sum wasn't the result of corporate bidders "losing their heads", but a careful strategy designed to maximise proceeds for the Treasury.
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The Original 5 Numbers:

Programme 1: Zero
What's 2 minus 2? The answer is obvious, right? But not if you wore a tunic, no socks and lived in Ancient Greece. For strange as it sounds, 'nothing' had to be invented, and then it took thousands of years to catch on.
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Programme 2: Pi
At its simplest, Pi is the ratio of the circumference of a circle to its diameter. At its most complex, it is an irrational number that cannot be expressed as the ratio of two whole numbers and has a random decimal string of infinite length.
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Programme 3: The Golden Ratio
Divide any number in the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, 55) by the one before it and the answer is always close to 1.618 the Golden Ratio.
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Programme 4: The Imaginary 'i'
The imaginary number takes mathematics to another dimension. It was discovered in sixteenth century Italy at a time when being a mathematician was akin to being a modern day rock star. The puzzle of the day was: "If the square root of +1 is both +1 and -1, then what is the square root of -1?"
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Programme 5: Infinity
Given the old maxim about an infinite number of monkeys and typewriters, one can assume that said simian digits will type up the following line from Hamlet an infinite number of times:
"I could confine myself to a nutshell and declare myself king of infinity".
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5 Numbers Quiz:
What number are you?

So you've read about five of the greatest numbers in the history of the world ever. But which number are you? Play our number game to reveal secrets about your inner self that you never dreamed of.
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The Number Four
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Game Theory
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