|
n
2
+ 1 > 2n
Similarly
(n
2
+ 1) - (n
2
- 1) = 2
therefore
n
2
+ 1 > n
2
- 1
This means that
n
2
+ 1
is the longest side of a triangle with sides that can be
written in the form
n
2
+ 1, n
2
- 1
and
2n
(where
n > 1
).
This can also be shown by means of the following graph (but this doesn't prove
anything):
According to Pythagoras's theorem, if the sum of the squares of the two shorter
sides equals the square of the hypotenuse, then the triangle is right-angled.
Therefore to prove that the triangle is right-angled we need to show that this is
the case.
The sum of the squares of the shorter two sides is
(
n
2
- 1)
2
+ (2
n
)
2
(
n
2
- 1)
2
+ (2
n
)
2
= n
4
- 2
n
2
+ 1
+ 4
n
2
= n
4
+ 2
n
2
+ 1
The square of the hypotenuse is
(
n
2
+ 1)
2
(
n
2
+ 1)
2
= n
4
+ 2
n
2
+ 1
Therefore the sum of the squares of the shorter two sides is equal to the square of
the hypotenuse and the triangle is right-angled.
And the converse of “A triangle with sides that can be written in the form
n
2
+ 1,
n
2
- 1
and
2
n
(where
n
> 1
) is right-angled” is “A triangle that is right-angled
has sides whose lengths can be written in the form
n
2
+ 1,
n
2
- 1
and
2
n
(where
n
> 1
).”
And a counterexample means finding a triangle which is right-angled but whose
sides cannot be written in the form
n
2
+ 1,
n
2
- 1
and
2
n
(where
n
> 1
).
So let the hypotenuse of the right-angled triangle
ABC
be
AB.
and let
AB = 65
and let
BC = 60
Then
CA = √ (AB
2
- BC
2
)
= √ (65
2
- 60
2
) = √ (4225 - 3600) = √ 625 = 25
Let
AB =
n
2
+ 1 = 65
then
n = √ (65 - 1) = √ 64 = 8
therefore
(
n
2
- 1) = 64 - 1 = 63 ≠ BC = 60 ≠ CA = 25
and
2n = 16 ≠ BC = 60 ≠ CA = 25
Therefore the triangle
ABC
is right-angled but it does not have sides which can
be written in the form
n
2
+ 1,
n
2
- 1
and
2
n
(where
n
> 1
).
QED
FIRST VINTAGE BOOKS EDITION, JULY 2008
All rights reserved. Published in the United States by Vintage Books, a division
of Random House, Inc., New York, and in Canada by Random House of Canada
Limited, Toronto. Originally published in hardcover in the United States by
Pantheon Books, a division of Random House, Inc., New York, in 2007.
Vintage and colophon are registered trademarks of Random House, Inc.
This book is a work of fiction. Names, characters, businesses, organizations,
places, events, and incidents either are the product of the author’s imagination or
are used fictitiously. Any resemblance to actual persons, living or dead, events,
or locales is entirely coincidental.
Underground logo, fabric designs, and line diagrams are reproduced with the
kind permission of Transport for London. Kuoni advertisement reproduced with
the kind permission of Kuoni Travel Ltd. A-level maths question reproduced
with the kind permission of Oxford Cambridge and RSA Examinations (OCR).
Every effort has been made to trace other copyright holders, and the publishers
will be happy to correct mistakes or omissions in future editions.
Library of Congress Cataloging-in-Publication Data
Haddon, Mark.
The curious incident of the dog in the night-time : a novel / Mark Haddon.—1st
ed.
p. cm.
Despite his overwhelming fear of interacting with people, Christopher, a
mathematically gifted, autistic fifteen-year-old boy, decides to investigate the
murder of a neighbor’s dog and uncovers secret information about his mother.
[1. Autism—Fiction. 2. Savants (Savant syndrome)—Fiction. 3. England—
Fiction.] I. Title.
PZ7.H1165 Cu 2003
[Fic]—dc21 2002031355
eISBN: 978-1-4000-7907-0
Copyright © 2002 by Mark Haddon
www.vintagebooks.com
v1.0
1
I found this in a book when Mother took me into the library in town in
1996.
Return to text.
2
This is not a
metaphor,
it is a
simile,
which means that it really did look like
there were two very small mice hiding in his nostrils, and if you make a picture
in your head of a man with two very small mice hiding in his nostrils, you will
know what the police inspector looked like. And a simile is not a lie, unless it is
a bad simile.
Return to text.
3
But I wouldn't have Shreddies and tea because they are both brown.
Return to text.
4
Once I didn't talk to anyone for 5 weeks.
Return to text.
5
When I was 6 Mother used to get me to drink strawberry-flavored slimming
meals out of a measuring jug and we would have competitions to see how fast I
could drink a quarter of a liter.
Return to text.
6
People say that you always have to tell the truth. But they do not mean this
because you are not allowed to tell old people that they are old and you are not
allowed to tell people if they smell funny or if a grown-up has made a fart. And
you are not allowed to say “I don't like you” unless that person has been horrible
to you.
Return to text.
7
Stupid things are things like emptying a jar of peanut butter onto the table in
the kitchen and making it level with a knife so it covers all the table right to the
edges, or burning things on the gas stove to see what happened to them, like my
shoes or silver foil or sugar.
Return to text.
8
I only did this once by borrowing the keys when she went into town on the
bus, and I hadn't driven a car before and I was 8 years old and 5 months so I
drove it into the wall, and the car isn't there anymore because Mother is dead.
Return to text.
9
It is permitted to move the chairs and the table in the kitchen because that is
different, but it makes me feel dizzy and sick if someone has moved the sofa and
the chairs around in the living room or the dining room. Mother used to do this
when she did the hoovering, so I made a special plan of where all the furniture
was meant to be and did measurements and I put everything back in its proper
place afterward and then I felt better. But since Mother died Father hasn't done
any hoovering, so that is OK. And Mrs. Shears did the hoovering once but I did
groaning and she shouted at Father and she never did it again.
Return to text.
12
This is really true because I asked Siobhan what people thought about
when they looked at things, and this is what she said.
Return to text.
Document Outline - Title Page
- Dedication
- Part One
- Appendix
- Copyright Page
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