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#1

# How To See in 4-D

We live in a world of three dimensions – up and down, side to side, and backwards and forwards, or any other three directions that are at right angles to each other. It’s easy to imagine something in one dimension, such as a straight line, or two dimensions, such as a square drawn on a sheet of paper. But how can we possibly learn to see in an extra dimension to those we’re familiar with? Where is this additional direction that’s perpendicular to the three we know?

#2

# Chance Is a Fine Thing

Thoroughly shuffle a deck of cards and the chances are that you’ve just done something unique. Almost certainly, no one in the history of the world has ever come up with the deck arranged in that particular order before. The reason’s simple: 52 different cards can be arranged in 52 × 51 × 50 × 49 × … × 3 × 2 × 1 ways. That’s a grand total of about 8 × 1067, or 80 million trillion trillion trillion trillion trillion different orderings of the cards.

#3

# Patterns At the Brink of Chaos

“How Long Is the Coast of Britain?” That’s part of the title of a paper by the Polish-born French-American mathematician Benoît Mandelbrot, a theorist at the IBM Thomas J. Watson Research Center, published in the journal Science in 1967. Strangely enough, there’s no definite answer because it depends on what scale is used.

#4

# Turing’s Fantastic Machine

In 1928, the German mathematician David Hilbert, renowned for challenging his peers with unsolved questions, posed what he called the Entscheidungsproblem or “decision problem”. This asked whether it’s always possible to find a step-by-step procedure to decide, in a finite time, if a given mathematical statement is true or not. Hilbert thought the answer would turn out to be “yes” but, in less than a decade, that hope had been dashed.

#5

# What Would Alien Music Be Like?

As they head toward the stars, Voyager 1 and 2 are carrying 90 minutes of music from different ages and regions of the world, including passages from Stravinsky’s “The Rite of Spring”, a gamelan piece from Indonesia, Bach’s Brandenburg Concerto No. 2, and Chuck Berry’s “Johnny B. Goode”. If aliens ever found one of the Golden Disks and managed to play the music as intended, would they’d recognise it for what it is. And, if alien music somehow reached our ears, would we appreciate it as being musical?

#6

# Prime Mysteries

A prime number is just a natural number that can be divided, without a remainder, by only itself and 1. This might not seem like a particularly special quality, but prime numbers occupy a position of central importance in maths. It isn’t an exaggeration to say that some of the greatest unsolved mysteries in the subject involve primes and that, on a practical level, these numbers play an important part in our daily lives.

#7

# Can Chess Be Solved?

Imagine an incredibly powerful computer that could always figure out the best move in any given possible chess position. ‘Best move’ means the one that leads most quickly to winning or, at the very least, not losing – in other words, the optimal outcome for the player. Now, suppose that this computer played against another that was identical to it in every respect. Which computer would win, or would it always be a draw?

#8

# What Is and What Should Never Be

The word ‘paradox’ comes from the Greek para (‘beyond’) and doxa (‘opinion’ or ‘belief’). Literally, then, it means anything that’s hard to believe or runs counter to our intuition or common sense. We’ll often say, in everyday conversation, that something is paradoxical just because it’s seems almost unbelievable. For example, the fact that in a room of 23 people there’s a 50/50 chance of two people having the same birthday is sometimes called the ‘birthday paradox’, even though it’s an easily-proven statistical fact and only surprising because it jars with our expectations.

#9

# You Can’t Get There From Here

Does space stop somewhere? Was there a beginning to time and will it ever end? Is there a biggest number? Even as children we ask these questions. Everyone, it seems, at one time or another, is curious about infinity.

#10

# The Biggest Number of All

Ask a child what’s the biggest number they can think of and the answer often runs along the lines of “50 thousand million billion trillion trillion…” until they run out of breath, with the odd nebulous ‘kazillion’ or ‘bazillion’ thrown in for good measure. Such numbers can certainly be big by everyday standards – maybe more than all the living things on Earth or all the stars in the universe. But they’re peanuts compared with the kinds of mind-bogglingly huge numbers that mathematicians can come up with.

#11

# Bend it, Stretch It, Any Way You Want To

Take two prints of the same picture. Put one of them down flat on a table, then crumple up the other, any way you like, providing you don't actually tear it, and place it somewhere on top of the uncrumpled print. It’s an inescapable fact that at least one point on the crumpled copy will lie directly over the corresponding point on the flat picture. You’ve just crossed over into the Topology Zone.

#12

# God, Gödel, and the Search for Proof

In about 1941, the Austrian-born logician Kurt Gödel, close friend of Albert Einstein at the Institute for Advanced Study in Princeton, proved that God exists. Unlike Einstein, who hovered between agnosticism and pantheism and once said he believed in “Spinoza’s god”, Gödel was a non-church-going theist who, according to his wife, “read the Bible in bed every Sunday morning”. The proof he published about the existence of God, however, had nothing to do with his Lutheran roots or anything that might ring a bell with ordinary folk. It was very much a product of his intellectually-lofty, mathematical mind.

#13

# The Maths Behind the World

In terms of intellectual ability, Homo sapiens hasn’t changed much, if at all, over the past 100,000 years. Put children from the time when woolly rhinos and mastodons still roamed the Earth into a present-day school and they would develop just as well as typical twenty-first century youngsters. Their brains would assimilate arithmetic, geometry, and algebra. And, if they were so inclined, there’d be nothing to stop them delving deeper into the subject and someday perhaps becoming professors of maths at Cambridge or Harvard.