It is not often that deep or interesting mathematics shows up in literary works. The Curious Incident of the Dog in the Night-time is not only a truly remarkable debut for its author, but it is also succeeds amazingly well at interweaving mathematics and mathematical modes of thought into a gripping storyline that is surprisingly passionate in its relentlessly analytical tone.
This mystery story is told in the first person by a 15-year-old boy with Asperger's Syndrome (which is essentially high functioning autism) whose obsession lies with mathematics. Christopher Boone is a mathematical savant, able to understand and create complex mathematical arguments with utmost ease. Though he is completely devoid of the ability to recognize human emotions, to comprehend metaphors, or filter the overwhelming sensory cacophony of the everyday world, Christopher sets out on a difficult journey, with mathematics serving as the only stable unchanging facet in his crumbling world.
There are already many excellent reviews of this book from a literary standpoint, so we shall content ourselves to address the mathematical aspects of this book. Since the narrator is both unyieldingly analytical and obsessed with mathematics, it is natural for him to cast his attempts in apprehend the bewildering (to him) chaos of our world in mathematical terms. For instance, on p. 100, Christopher contemplates how mysterious-looking situations can sometimes be explained with simple mathematical rules. He gives, as an example for the reader, a clear and understandable description of the behavior arising from a logistic equation that expresses the population of a system of animals as Nnew = kNold (1 - Nold). Varying the parameter k can give several different outcomes, including extinction, stable growth to a limiting population, cyclical oscillation of population sizes, and even chaotic variation. This example, originating in work of May, Oster, and Yorke, was one of the early manifestations of chaos theory. More information can be found, for example, in an essay by Robert May. There are of course a vast range of books on this topic; we shall mention only Nonlinear Dynamics and Chaos, by Steven Strogatz, which includes a detailed discussion of this example in section 10.2. Christopher concludes this example by discussing how it helps him to make sense of the world:
And it means that sometimes a whole population of frogs, or worms, or people, can die for no reason whatsoever, just because that is the way the numbers work.
Prime numbers are also a recurrent theme in the book. Christopher loves primes; he knows each one up to 7,057 and numbers all of this book's chapters with prime numbers. He introduces prime numbers to the reader with a very readable explanation of the Sieve of Eratosthenes, and he makes it clear to even the most uninitiated reader that finding large primes is quite difficult. He also alludes to the Prime Number Theorem (or, rather, his inability to recall it when over-stressed) on p. 212. Interested readers can find a statement of the theorem (conjectured by Gauss and proven by Hadamard and de la Vallée Pouisson) in many places, including http://mathworld.wolfram.com/PrimeNumberTheorem.html. Christopher also displays an amusing blend of precocity and boyish cloak-and-dagger fantasy when he says (on p. 12),
Prime numbers are useful for writing codes and in America they are classed as Military Material and if you find one over 100 digits long you would have to tell the CIA and they buy it off you for $10,000.
This is of course an allusion to the use of large primes in public-key encryption systems such as the Rivest-Shamir-Adleman (RSA) system. This system is based on the empirical difficulty of factoring the product of two very large primes, and forms the basis of secure cryptography. (Cryptography has widespread applications in military and commercial settings, including the familiar SSL protocol, which is widely used in securing Internet commerce transactions.) In connection with this, we mention a recent breakthrough that has been made in this area by Agrawal, Kayal, and Saxena, establishing a polynomial-time algorithm for determining whether a given number is prime. (See the November 2002 issue of FOCUS for more details.)
Perhaps the most amusing mathematical interlude of the novel is the discussion of the Monty Hall Problem on p. 64, which Christopher gives as an example of how one can use mathematics to grasp and analyze the uncertain outcomes of life's events. Christopher mentions Marilyn vos Savant's infamous column in Parade magazine, along with a rather embarrassing slew of snide letters written by professional mathematicians excoriating Marilyn for publishing her (ultimately correct) solution. It is interesting that Mark Haddon chooses to provide both a theoretical solution to the problem and a visual way of figuring out the correct answer to this conditional probability conundrum. The visual presentation is so transparent that it could be a wonderful way to explain the Monty Hall problem in the introductory probability and statistics courses. However, we should point out that the most crucial subtlety is not mentioned in the discussion, namely that the answer depends on Monty's strategy. If you select a door with no prize, and you don't know that Monty will never open the door with the prize in it when opening the second door, then his action of opening an empty second door will not give you any information. But if you know he will never open the door with a prize in it (i.e., P(HZ|CY) = P(HY|CZ) = 1 in Christopher's notation) then Monty's action of opening an empty door indeed provides some additional information to you. For a much fuller discussion of the numerous variants of this problem, see the article The WWW Tackles the Monty Hall Problem.
Other deductive reasoning techniques, such as Occam's razor and enumeration of cases, also find their place in the novel. And the appendix contains a proof that a triangle with sides n2 + 1, n2 - 1 and 2n (for n > 1) is a right triangle. (This is an immediate consequence of the Pythagorean theorem.) It also shows that the converse of the theorem is false. This is probably as close as a mathematical proof ever gets to the best seller list of the New York Times.
In Christopher's self-analysis of his inability to understand humor, he dissects a joke's punch line as follows (p. 8):
This will not be a funny book. I cannot tell jokes because I do not understand them. Here is a joke, as an example. It is one of father's. His face was drawn but the curtains were real. I know why this is meant to be funny. I asked. It is because drawn has three meanings, and they are (1) drawn with a pencil, (2) exhausted, and (3) pulled across a window, and meaning 1 refers to both the face and the curtains, meaning 2 refers only to the face and meaning 3 refers only to the curtains..."
This analysis echoes the theme of a delightful book by John Allen Paulos (of Innumeracy fame) entitled Mathematics and Humor. He sets forth a theory of humor in which the set-up statements of a joke are intended to make the listener believe that the joke's statements are being made with respect to a certain axiom system. The punch line, however, makes it clear that the joke's statements are supposed to be interpreted with respect to a different axiom system. Paulos surmises, roughly, that our sensation of humor arises from our perception of this sudden shift in the axiom system. He also discusses an analogy between humor and catastrophe theory.
Another interesting mathematical topic that makes an appearance (when Christopher is trying to take his mind off of the overwhelming stimuli by retreating into a mental solitaire game) is a game that he calls Conway's Soldiers. In this game we have an infinite chessboard, and every square below a certain horizontal line has a tile on it. A tile is allowed to move by jumping horizontally or vertically over another tile into an empty space, and the jumped-over tile is removed. It is a remarkable fact that, regardless of what strategy is employed, no soldier can ever go more than 4 rows above the line. One very elegant proof of this fact involves constructing a numerical labeling of the possible tile positions that behaves in a certain way with respect to the jumps, and showing that there is no way to attain the label that is sitting 5 rows above the line. This proof can be found in Chapter 23 of the monumental multi-volume compendium Winning Ways for Your Mathematical Plays, by Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy, which contains an enormous wealth of material on mathematical aspects of all sorts of games. A short article on Conway's Soldiers can also be found in Chapter 3 of the MAA's own Mathematical Gems II, by Ross Honsberger.
Christopher lays bare the relentlessly logical chains of reasoning that underlie his outwardly bizarre behaviors with a guileless inexorability that sweeps the reader into his realm. I think that most readers, following Christopher's travails as he struggles with navigating the London subway system, will follow along so closely that they will even empathize with Christopher as he threatens passersby with his Swiss Army knife. Exposing the workings of his mind is, in many respects, similar to an exposition of the thought processes involved in mathematical deduction (a wholly different process, we should hasten to add, from that of mathematical discovery). As Christopher reaches a great crisis (p. 130), he enumerates the possible choices graphically, and then proceeds to eliminate them. I suspect that many mathematicians will empathize with the manner in which Christopher approaches decisions, and some may even see some facets of their own personalities in him.
The Curious Incident of the Dog in the Night-time is a book that is hard to put down. It brilliantly pulls the reader inside the brain of an autistic person, with a spine-tingling effect similar to that of Oliver Sack's The Man Who Mistook His Wife for a Hat. One feels compelled to find out where Christopher's detective work will take him. And the mathematical lanes make this story of compassion and of hope all the more veritable and richer. It is the reviewer's hope that this book will whet the mathematical appetite of some curious readers, and that the references herein may provide a guide for further exploration.
Maria G. Fung (firstname.lastname@example.org) is an assistant professor of mathematics at Western Oregon University. Her interests are the mathematical preparation of K-8 teachers and the representation theory of Lie groups.