Daniel the Mathemagician
March 24, 2007
Hello everyone,
Today I will share with you the secrets of being a mathemagician, including how to count ghosts. OoOooOooOOooohhhh… For you non-non math or ghost people, I’ll give a cultural update soon too.
I had two midterms this week and, thankfully, survived both of them. My Combinatorics midterm was extremely hard—everyone thought they did poorly, so we all failed together. I think I did reasonably well on my Geometry midterm.
In Geometry we are done with Euclidean Geometry (transformations, symmetry groups, hyperplanes, etc) and we started studying spherical geometry—a land where the angles of a triangle don’t add up to 180 degrees and where the shortest distance between two points is a curve. The implications of spherical geometry don’t make sense unless you think of it this way: imagine you are some creature living on a sphere, though you don’t know it’s a sphere; the sphere is flat from your perspective (perhaps you are living in Europe in the Middle Ages). As such, you can only use empirical evidence from living on the surface of the sphere to describe the land. It becomes very philosophical: “what is a line”, “how do you describe a line”? Anyway, that was our 15 minute introduction to spherical geometry after our midterm.
What do I do in my math classes? Simply speaking, I learn about numbers in Number Theory, count these numbers in Combinatorics and study shapes in Geometry. Unfortunately, it’s not that easy (and most of the time we don’t even use numbers), nor can these disciplines be described using one word. What do mathematicians do in these disciplines (you may ask)? Solve problems. How? By proving things. But, what is a proof? A logical argument for why a conjecture is true. By abstracting problems, using theorems and definitions and experimenting with simpler cases, mathematicians can prove conjectures that seem impossible to prove. That’s why I am a mathemagician; I do the seemingly impossible. There have been many mathemagicians before me working on these same proofs. They, however, have discovered some sort of trick for proving the seemingly impossible problem. So as students, we are in search for these tricks—these magical, yet logical, crucial steps for constructing a proof. Without the the “right” tricks, a problem becomes much more difficult—if not impossible. Unfortunately, these tricks can be just as hard to find, and a good mathemagician doesn’t reveal his secrets (at least when he or she is your professor). And there are many tricks yet to be discovered.
So how do we find tricks? In the BSM program, our professors encourage us to try the following procedure for solving a problem: Experiment, Conjecture, Prove. This may sound obvious, but many times we young mathemagicians are eager to jump straight into the proving step (and usually fail). In the experimenting step, we try out simpler cases of the problem and alter the problem to see if we notice any patterns. If we find a pattern, then we make a conjecture that generalizes this observation. Then we must prove this conjecture to be true.
Let me give you an example. In Combinatorics one of our easier homework problems last week was the following (I’ll take out the math-ese): Say you have n consecutive numbers in a line (1,2,3,…,n). How many ways can you pick numbers and put them into your pocket such that there are no consecutive numbers in your pocket (and the order you pick the numbers does not matter). If you try to make a conjecture too quickly from this information, this problem is nearly impossible to solve (or you need a very abstract, flexible mind to do so). However, if you first experiment with this problem for small values of n, the pattern becomes obvious—the Fibonacci numbers (1,1,2,3,5,8,…), where 1 is the first and second number of the sequence and you add those two numbers together to get the next number, and so on. Through experimenting, our conjecture is that if you have n original numbers, the number of different pockets (or ways you can put the numbers into a single pocket) is the n+1 number in the Fibonacci sequence. The answer grows really fast, for if you have 30 numbers, then you have 1,346,269 different pockets. That is, if no two pockets can contain the same numbers, nearly everyone in Manhattan has different pockets full of numbers! Now based on that information, you then have to come up with a proof of why the Fibonacci numbers makes sense (i.e. why is this conjecture true). I won’t go into that because it might be too mathy and this email is getting long.
Every week I have 15 such proofs to write as homework for my three math classes. One proof can take anywhere from 20 minutes to over six hours to complete. Luckily my assignments are due on different days of the week, though I’ve realized that I must start these problems plenty of days in advance to let thoughts percolate. I also have a study group in each class where we give each other hints or work through problems together if we are stuck. I am realizing that there is a fine art to writing a good proof. Writing a logical, coherent proof is almost as hard as solving the proof itself.
Let me share one last trick: the trick of a bijection (i.e. one-to-one correspondence), which I have used in every upper-level math course so far. I won’t give you a mathematical definition, but I will give you an example. Say your professor assigns you to count the number of ghosts in a graveyard. Well, assuming you are human, you can’t see ghosts. Instead, you count the number of gravestones. This is true with the assumption there is a one-to-one correspondence between ghosts and gravestones, and there is a ghostly barrier preventing any wandering spirits from entering or escaping the graveyard. The function that maps each ghost to its respective gravestone is a bijection. Interestingly enough, real life bijections can be much more abstract than counting ghosts.
Moral: you can be a mathemagician too. Next time you are in a graveyard or at a funeral, tell your friends that you can count how many ghosts are present.
Have a good day,
Daniel the Mathemagician