Professor Stewart's Hoard of Mathematical Treasures Read Online Free Page B
Author: Ian Stewart
Tags: General, Mathematics
not rigorous, because the numbers arising are not truly random. Exceptions might still occur anyway, and even if the argument were rigorous it would not rule out running into a different cycle.
If the process is extended so that it can start with zero or negative integers, four other cycles appear. They all involve numbers bigger than -20, so you might like to search for them (see the answer on page 279). The conjecture now becomes: these five cycles are all that can happen.
There are also connections with chaotic dynamics and fractal geometry, which lead to some beautiful ideas and pictures, but don’t solve the problem either. There’s a lot of information about this problem on the internet, for example:
en.wikipedia.org/wiki/Collatz_conjecture
mathworld.wolfram.com/CollatzProblem.html
www.numbertheory.org/3x+1/
The Jeweller’s Dilemma
Rattler’s Jewellers had promised Mrs Jones that they would fit her nine pieces of gold chain together to make a necklace, an endless loop of chain. It would cost them £1 to cut each link, and £2 to rejoin it - a total of £3 per link. If they cut one link at the end of each separate piece, linking the pieces one at a time, the total cost would be £27. However, they had promised to do this for less than the cost of a new chain, which was £26. Help Rattler’s avoid losing money - and, more importantly, make the cost to Mrs Jones as small as possible - by finding a better way to fit the pieces of chain together.
Nine lengths of chain.
Answer on page 279
What Seamus Didn’t Know
Our first cat, who rejoiced in the name Seamus Android, was possibly one of the few cats on earth that did not always land on its feet. Seamus didn’t have a clue. He would come down the stairs one step at a time, head first. At one point, Avril tried to train him to land on his feet by holding him upside down over a thick cushion and letting go. He liked the game but made no effort to turn in mid-air.
Oops . . . What do I do now?
There is a mathematical issue here. Associated with any moving body is a quantity called angular momentum, which, roughly speaking, is the mass multiplied by rate of spin about a suitable axis. Newton’s laws of motion imply that the angular momentum of any moving body is conserved, that is, does not change. So how can a falling cat turn over without touching anything?
Answer on page 279
Why Toast Always Falls Buttered-Side Down
A cat is not the only proverbial falling object. Toast is another. It always lands buttered-side down. If not, you must have buttered the wrong side.
Curiously, there is some truth to this adage. Robert Matthews has analysed the dynamics of falling toast, which does in fact have a propensity to land in a way that gets butter (or in my case marmalade) all over the carpet and ruins the toast. This lends support to Murphy’s law: Anything that can go wrong, will go wrong.
Matthews applied some basic mechanics to explain why toast tends to land buttered-side down. It turns out that tables are just the right height for the toast to make one half turn before it hits the floor. This may not be an accident, because the height of tables is related to the height of humans, and if we were much taller then the force of gravity would smash our skulls if we tripped. Matthews thus traces the trajectory of toast to a universal feature of the fundamental constants of the universe in relation to intelligent life forms. To my mind, this is probably the most convincing example of ‘cosmological fine-tuning’.
The Buttered Cat Paradox
Suppose we put the previous two pieces of folklore together:
• Cats always land on their feet.
• Toast always lands buttered-side down.
Therefore . . . what? The buttered cat paradox takes these statements as given, and asks what would happen to a cat, dropped from a considerable height, to whose back is firmly attached a slice of buttered toast - buttered-side outwards from the cat, of course. 4
At the time of
If the process is extended so that it can start with zero or negative integers, four other cycles appear. They all involve numbers bigger than -20, so you might like to search for them (see the answer on page 279). The conjecture now becomes: these five cycles are all that can happen.
There are also connections with chaotic dynamics and fractal geometry, which lead to some beautiful ideas and pictures, but don’t solve the problem either. There’s a lot of information about this problem on the internet, for example:
en.wikipedia.org/wiki/Collatz_conjecture
mathworld.wolfram.com/CollatzProblem.html
www.numbertheory.org/3x+1/
The Jeweller’s Dilemma
Rattler’s Jewellers had promised Mrs Jones that they would fit her nine pieces of gold chain together to make a necklace, an endless loop of chain. It would cost them £1 to cut each link, and £2 to rejoin it - a total of £3 per link. If they cut one link at the end of each separate piece, linking the pieces one at a time, the total cost would be £27. However, they had promised to do this for less than the cost of a new chain, which was £26. Help Rattler’s avoid losing money - and, more importantly, make the cost to Mrs Jones as small as possible - by finding a better way to fit the pieces of chain together.
Nine lengths of chain.
Answer on page 279
What Seamus Didn’t Know
Our first cat, who rejoiced in the name Seamus Android, was possibly one of the few cats on earth that did not always land on its feet. Seamus didn’t have a clue. He would come down the stairs one step at a time, head first. At one point, Avril tried to train him to land on his feet by holding him upside down over a thick cushion and letting go. He liked the game but made no effort to turn in mid-air.
Oops . . . What do I do now?
There is a mathematical issue here. Associated with any moving body is a quantity called angular momentum, which, roughly speaking, is the mass multiplied by rate of spin about a suitable axis. Newton’s laws of motion imply that the angular momentum of any moving body is conserved, that is, does not change. So how can a falling cat turn over without touching anything?
Answer on page 279
Why Toast Always Falls Buttered-Side Down
A cat is not the only proverbial falling object. Toast is another. It always lands buttered-side down. If not, you must have buttered the wrong side.
Curiously, there is some truth to this adage. Robert Matthews has analysed the dynamics of falling toast, which does in fact have a propensity to land in a way that gets butter (or in my case marmalade) all over the carpet and ruins the toast. This lends support to Murphy’s law: Anything that can go wrong, will go wrong.
Matthews applied some basic mechanics to explain why toast tends to land buttered-side down. It turns out that tables are just the right height for the toast to make one half turn before it hits the floor. This may not be an accident, because the height of tables is related to the height of humans, and if we were much taller then the force of gravity would smash our skulls if we tripped. Matthews thus traces the trajectory of toast to a universal feature of the fundamental constants of the universe in relation to intelligent life forms. To my mind, this is probably the most convincing example of ‘cosmological fine-tuning’.
The Buttered Cat Paradox
Suppose we put the previous two pieces of folklore together:
• Cats always land on their feet.
• Toast always lands buttered-side down.
Therefore . . . what? The buttered cat paradox takes these statements as given, and asks what would happen to a cat, dropped from a considerable height, to whose back is firmly attached a slice of buttered toast - buttered-side outwards from the cat, of course. 4
At the time of
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