The Mathematics of
“the curious incident of the dog
in the night-time”
by Mark Haddon
D.N. Seppala-Holtzman
St. Joseph’s College
faculty.sjcny.edu/~holtzman
Christopher Boone
► The
protagonist of this novel is Christopher.
► He is a teenager living in Swindon, England.
► He is suffering with an un-named disorder
most likely to be Asperger’s Syndrome.
► He announces himself to be 15 years, 3
months and 2 days at the outset.
► With this, he declares his affinity with
precision.
Asperger’s Syndrome
► There
are many variations on this and related
disorders.
► These range from various forms of autism to a
host of mental disorders.
► Examples include the real person upon which the
film “Rain Man” was based (Kim Peek) and the
twin brothers who communicated with one
another by exchanging references to large prime
numbers. (See “The Man Who Mistook His Wife for
a Hat and Other Clinical Tales” by Oliver Sacks.)
Mathematics and Mental Disorders
► There
appears to be a correlation between
certain types of mental disorders and an
affinity for or facility with mathematics.
► Often the mathematics in question is
nothing more than computational abilities,
albeit on an extraordinary level.
► On occasion, it is on a higher plane.
Christopher & Math I
► Christopher
demonstrates his affinity with
mathematics in many ways.
► The chapters of the book are numbered by
primes.
► He states: “I think prime numbers are like life.
They are very logical but you could never work out
the rules…”
► Christopher may not know it but one of the most
important problems in mathematics, today, is to
“work out the rules.” This is the Riemann
Hypothesis.
Christopher & Math II
► Christopher
seems to find solace in the
precision of mathematics as it contrasts with
the vagaries of normal human interaction.
► He seems unable to supply the tacit,
assumed information that the un-afflicted
see as obvious.
► Keep off the grass: which grass?
► Be quiet: for how long?
Christopher & Math III
► Christopher
finds mental calculations to be
both easy and calming.
► He finds producing a list of powers of 2 to
be therapeutic.
► The one and only joke that he both
understands and appreciates (regarding
brown cows in Scotland) has, at its core, the
issues of assumptions and precision.
Christopher & Math IV
► Christopher
sees himself as a computer, a
machine extremely good in calculating but
notoriously poor in supplying assumed
information.
► When overwhelmed, he wishes that he
could just press “CTRL + ALT+DEL.”
Mathematical Examples from the
Book
► There
are many explicit examples of “real”
mathematics, not just computations, that
Christopher explores in this book, including:
► Chaos
► The Monty Hall Problem
► Conway’s Soldiers
► Pythagorean triples
► Let us have a closer look at each of these.
Chaos I
► Christopher
tells us that he has been
reading the book “Chaos” by James Gleick.
► He relates the treatment in this book of
“the” logistics equation to the population of
frogs in a pond.
► I put the word “the” in quotes as there are
many such equations. A more sophisticated
version is actually a differential equation.
Chaos II
► The
equation used here is this:
Nnew  r ( Nold ) * (1  Nold )
► This
relates the population of the next
generation to that of the previous one.
Chaos III
► For
simplicity’s sake, we treat the
population as a number between 0 and 1.
We could think of this as 0% and 100% of
the possible populations that a given
environment could support.
► Note that there is a parameter, r, which
gives the net “growth” rate (growth being,
possibly, negative).
Chaos IV
► Chaos
is the study of dynamical systems
where the state of the system at a given
time is a function of the state at a previous
time.
► Sometimes, even very simple rules (like the
logistics equation), can produce
unpredictable, chaotic, results.
► Hence the name, Chaos Theory.
Chaos V
► It
turns out that changing the parameter, r,
in a certain range yields highly predictable
results.
► When a certain threshold for r is crossed,
the behavior changes to more complex but,
nonetheless, predictable.
► When another threshold is crossed, the
behavior goes wild: Chaos ensues!
Chaos VI
► To
examine this behavior, we shall use
graphical analysis.
► We shall draw the curve given by the
equation:
► y = r*x*(1-x)
► We will also draw the line y = x.
► To examine iterative behavior, we shall
trace the trajectory of a beginning value.
Chaos VII
► Start
with an initial value on the x-axis.
► Draw a vertical line from here to the curve. The ycoordinate of this point gives the “output” for
given initial “input.”
► Now draw the horizontal line from this point to the
line y = x. This makes the new x-value (the new
input) the output from the previous iteration.
► Repeating this process gives the long-term
behavior of the system.
Chaos VIII r = 2.2 (fixed point)
Chaos IX r = 3.0 (period 2)
Chaos X r= 3.6 (period 8)
Chaos XI r = 4.0 (Chaos)
Chaos XII
► This
is what Christopher was describing
when he was discussing the frog pond at his
school.
► His point was that life is more chaotic than
we sometimes think.
► “That is the way the numbers work,” he
says.
Monty Hall I
► Another
topic that Christopher brings up is the
famous Monty Hall problem.
► Named for the game show host, the problem is
this: Suppose you were shown 3 doors. Behind
one was a car and behind the other two was a
goat. You select a door, say door #2.
► Now Monty Hall opens a different door, say #3 to
reveal a goat. He offers to let you change your
choice from #2 to the remaining door #1. What
should you do?
Monty Hall II
► Most
people believe that Monty Hall has
given you no new information and that you
might as well stay with your original choice.
► In fact, you go from a 1/3 chance of
winning to a 2/3 chance of winning by
switching.
► Here’s why:
Monty Hall III
► Initially,
you had a 1/3 chance of winning the car
with your door selection. You would win a goat
with probability 2/3.
► Now, if your initial choice was right, you would
lose the car by switching. But, if your initial choice
was wrong, you would win the car by switching.
► That is, switching doors turns an initial loss into a
win and an initial win into a loss.
► Thus, switching makes the likelihood of winning a
car 2/3.
Monty Hall IV
► This
problem was put to Marilyn vos Savant
in her Parade Magazine column.
► She answered it correctly but was roundly
criticized and scorned by many highly
educated people.
► Christopher enjoys the fact that the
numbers lead to a counter-intuitive result.
Conway’s Soldiers I
► This
is a problem that Christopher enjoys doing to
make his “head clearer.”
► Start with an infinite chessboard. Draw a
horizontal line through some row. Place markers
(soldiers) on all of the squares below this row.
► Now, if a marker can jump (horizontally or
vertically) over another marker and land on an
empty square, it may do so.
► The jumped over marker is then removed from the
board.
Conway’s Soldiers II
► The
object of the exercise is to advance
markers as far above the horizontal line as
possible.
► The game was named for John H. Conway,
a mathematician at Princeton (formally from
Cambridge). He proved that, no matter
how the game is played, it is impossible to
get beyond 4 rows above the line.
Conway’s Soldiers III
Conway’s Soldiers IV
► The
previous slide shows configurations that
allow advances of 1, 2, 3 and 4 rows.
► The problem has been generalized into
more than two dimensions.
Pythagorean Triples I
► We
end with a discussion of the A-levels
problem that Christopher was most proud of
solving:
► Let n be an integer > 1. Prove that a
triangle with sides n2 +1, n2 -1 and 2n is a
right triangle.
► Give a counter example to show that the
converse is false.
Pythagorean Triples II
►A
set of three positive integers which form the
sides of a right triangle are called a Pythagorean
triple.
► Most people know that 3 – 4 – 5 is a Pythagorean
triple.
► The problem asks one to prove that n2+1, n2-1
and 2n makes up a Pythagorean triple for any
integer n > 1 but that there are other Pythagorean
triples that are not of this form.
Pythagorean Triples III
► Christopher,
in his solution, spends quite a
bit of time showing that n2+1 must be the
length of the hypotenuse. This was unnecessary as showing that its square is the
sum of the squares of the other two lengths
forces it to be the longest side, i.e. the
hypotenuse.
Pythagorean Triples IV
► Ultimately,
Christopher’s proof came down
to the simple verifying calculation:
► (n2-1)2 +
► This
(2n)2 = (n2+1)2
makes this trio a Pythagorean triple.
Pythagorean Triples V
► To
prove that not all Pythagorean triples are of
this form, he mentally searched through triples
such that the sum of the squares of the two
smaller numbers equaled the square of the
largest, but failed to conform to the given pattern.
► He chose 25 – 60 - 65.
► Note that this could be found by multiplying each
term of the well-known 5 – 12 – 13 triple by 5.
Indeed, 5 – 12 – 13 would have been an easier
counter-example to come up with.
Pythagorean Triples VI
► What
Christopher, apparently, did not know is that
all primitive Pythagorean triples (i.e. relatively coprime) are of the following form:
► For s > t > 0: x = 2st y = s2-t2 z = s2+t2
► In addition, we ask that s and t be relatively prime
and not differ by a multiple of 2.
► Setting t = 1 and s = n, gives the form that
Christopher was asked to verify. Setting t >1 will
give a counter example.
Conclusion
► These
examples typify the thoughts that
occupied Christopher’s mind and, evidently,
provided him some modicum of solace.
► They offered him the harbor of an orderly
inner world where chaos ruled outside.