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: • 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 y-coordinate 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 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 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 un-necessary 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 + (2n)2 = (n2+1)2 • This 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 co-prime) 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.