Read Online Professor Stewart's Hoard of Mathematical Treasures by Ian Stewart - Free Book Online Page A
Authors: Ian Stewart
Tags: General, Mathematics
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7 times the previous one. The scribe gives a short cut:
2,801×7 = 19,607
Note that 2,801 = 1 + 7 + 49 + 343 + 2,401. These numbers are the first few powers of 7. I have no idea why the scribe thought it sensible to add up such diverse items, mind you.
Still on exponential growth: the Humane Association has pointed out that if two cats and their kittens breed for 10 years, with each cat having two litters of three surviving kittens per year, then the cat population grows like this:
12 66 382 2,201 12,680 73,041 420,715 2,423,316 13,968,290 80,399,780
In the 1960s the Russian mathematician Vladimir Arnold studied a map (another word for ‘function’ or ‘transformation’) from the torus to itself, defined by
( x, y ) → ( 2x + y , x + y ) (mod 1)
where x and y lie between 0 (included) and 1 (excluded), and (mod 1) means that everything before the decimal point (the integer part) is ignored. So 17.443 (mod 1) = 0.443, for instance. The dynamics of this map are chaotic (Cabinet, page 117); also, it ‘preserves area’, meaning that areas don’t change when it is applied. So it provided a simple model for more complicated area-preserving maps arising naturally in mechanics.
This map quickly became known as Arnold’s cat, because he illustrated its effect by drawing a cat on the torus, and showing how the cat distorts when the map is applied. The same thing is done with a picture of a real cat at:
upload.wikimedia.org/wikipedia/commons/a/a6/Arnold_cat.png
www.nbi.dk/CATS/PICS/cat_arnold.gif
Author Theoni Pappas wrote a children’s book, The Adventures of Penrose the Mathematical Cat, presumably named after mathematical physicist Roger Penrose.
Arnold’s cat.
In the book Mathematicians in Love by Rudy Rucker, two mathematics graduate students prove a theorem characterising all dynamical systems in terms of objects from Dr Seuss’s The Cat in the Hat. 16
In his 1964 research text Abelian Categories, Peter Freyd included the index entry ‘kittygory’. The page concerned refers to a ‘small category’.
There’s a mathematician named Nicholas Katz - does that count?
Um - Felix Hausdorff?
The Rule of Eleven
There’s an old test for divisibility by 11, seldom taught in these days of calculators. Suppose, for example, that the number is 4,375,327. Form the two sums
4 + 7 + 3 + 7 = 21, 3 + 5 + 2 = 10
formed by taking every alternate digit ( 4 3 7 5 3 2 7 ). Take the difference, 21-10 = 11. If this difference is exactly divisible by 11, so is the original number, and conversely. (The number 0 is exactly divisible by 11, being equal to 11×0.) Here the difference is 11 itself, which is divisible by 11, so the test says that 4,375,327 is divisible by 11. In fact, it is equal to 11×397,757. Initial zeros make no difference, by the way, since they add zero to whichever sum they appear in.
Here are two puzzles and a question; the puzzles are easier if you use this test.
• Find the largest number that uses each of the digits 0-9 exactly once, and is divisible by 11 without remainder.
• Find the smallest such number, not starting with 0.
• While we’re at it: what is the smallest positive multiple of 11 for which the test does not yield a difference of zero? Answers on page 290
Digital Multiplication
The square array uses each of the nine digits 1-9. The second row 384 is twice the first row 192, and the third row 576 is three times the first row.
1
9
2
3
8
4
5
7
6
There are three other ways to do this. Can you find them?
Answer on page 291
Common Knowledge
There is an entire genre of puzzles that rests on the counterintuitive properties of ‘common knowledge’ - something that has been made public, so that not only does everyone involved know it, but they know that everyone knows it, and they know that everyone knows that everyone knows it . . . A traditional case concerns the curious habits of the obscure but very polite Glaberine 17 order of monks.
Not ‘habits’ as in clothing, you
2,801×7 = 19,607
Note that 2,801 = 1 + 7 + 49 + 343 + 2,401. These numbers are the first few powers of 7. I have no idea why the scribe thought it sensible to add up such diverse items, mind you.
Still on exponential growth: the Humane Association has pointed out that if two cats and their kittens breed for 10 years, with each cat having two litters of three surviving kittens per year, then the cat population grows like this:
12 66 382 2,201 12,680 73,041 420,715 2,423,316 13,968,290 80,399,780
In the 1960s the Russian mathematician Vladimir Arnold studied a map (another word for ‘function’ or ‘transformation’) from the torus to itself, defined by
( x, y ) → ( 2x + y , x + y ) (mod 1)
where x and y lie between 0 (included) and 1 (excluded), and (mod 1) means that everything before the decimal point (the integer part) is ignored. So 17.443 (mod 1) = 0.443, for instance. The dynamics of this map are chaotic (Cabinet, page 117); also, it ‘preserves area’, meaning that areas don’t change when it is applied. So it provided a simple model for more complicated area-preserving maps arising naturally in mechanics.
This map quickly became known as Arnold’s cat, because he illustrated its effect by drawing a cat on the torus, and showing how the cat distorts when the map is applied. The same thing is done with a picture of a real cat at:
upload.wikimedia.org/wikipedia/commons/a/a6/Arnold_cat.png
www.nbi.dk/CATS/PICS/cat_arnold.gif
Author Theoni Pappas wrote a children’s book, The Adventures of Penrose the Mathematical Cat, presumably named after mathematical physicist Roger Penrose.
Arnold’s cat.
In the book Mathematicians in Love by Rudy Rucker, two mathematics graduate students prove a theorem characterising all dynamical systems in terms of objects from Dr Seuss’s The Cat in the Hat. 16
In his 1964 research text Abelian Categories, Peter Freyd included the index entry ‘kittygory’. The page concerned refers to a ‘small category’.
There’s a mathematician named Nicholas Katz - does that count?
Um - Felix Hausdorff?
The Rule of Eleven
There’s an old test for divisibility by 11, seldom taught in these days of calculators. Suppose, for example, that the number is 4,375,327. Form the two sums
4 + 7 + 3 + 7 = 21, 3 + 5 + 2 = 10
formed by taking every alternate digit ( 4 3 7 5 3 2 7 ). Take the difference, 21-10 = 11. If this difference is exactly divisible by 11, so is the original number, and conversely. (The number 0 is exactly divisible by 11, being equal to 11×0.) Here the difference is 11 itself, which is divisible by 11, so the test says that 4,375,327 is divisible by 11. In fact, it is equal to 11×397,757. Initial zeros make no difference, by the way, since they add zero to whichever sum they appear in.
Here are two puzzles and a question; the puzzles are easier if you use this test.
• Find the largest number that uses each of the digits 0-9 exactly once, and is divisible by 11 without remainder.
• Find the smallest such number, not starting with 0.
• While we’re at it: what is the smallest positive multiple of 11 for which the test does not yield a difference of zero? Answers on page 290
Digital Multiplication
The square array uses each of the nine digits 1-9. The second row 384 is twice the first row 192, and the third row 576 is three times the first row.
1
9
2
3
8
4
5
7
6
There are three other ways to do this. Can you find them?
Answer on page 291
Common Knowledge
There is an entire genre of puzzles that rests on the counterintuitive properties of ‘common knowledge’ - something that has been made public, so that not only does everyone involved know it, but they know that everyone knows it, and they know that everyone knows that everyone knows it . . . A traditional case concerns the curious habits of the obscure but very polite Glaberine 17 order of monks.
Not ‘habits’ as in clothing, you
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