Teachers As Scholars (Steven J Miller: sjm1 AT williams.edu)

"A-ha!" Moments in Mathematics and the Sciences

Monday January 13 and Monday February 3, 2014
Harvard Hillel, 52 Mount Auburn Street, Rabb Hall (1st Floor)

Videos and Links for Topics: Below are the videos of the main items I presented at the sessions, as well as some links and supporting materials by myself and the class; discussions among participants was not recorded. Feel free to share these resources with anyone you wish. These are meant to be just quick summaries to remind you of what we discussed; if there are items I can add to make these more useful, just let me know. I'm happy to work with you on creating units for your students (and if that means having me do new lectures and record and upload them from Williams, I'm happy to do that).

• HOMEWORK PROBLEMS:
• Let A win $$p\%$$ of its games, and B win $$q\%$$ of their games ($$q$$ is not necessarily $$1-p$$). Imagine A and B play each other now. Which of the following four formulas do you think does a good job of estimating the probability A beats B: $$(p \pm pq) / (p + q \pm 2pq)$$? Remember there are four choices for the two signs: both positive, one positive and one negative (two ways), both negative.
• Consider a $$5 \times 5$$ chess board. It is possible to place $$5$$ queens on the board in such a way that there are three squares they cannot attack, so that you can then safely place $$3$$ pawns on those squares. How can you do this?
• Watch the YouTube video (http://www.youtube.com/watch?v=Esa2TYwDmwA) on Difference Equations and Roulette. Let me know if something like this could be of use to you. What can I do to make videos like this more useful? I can make several of these with OIT here, but input on what would be useful to you would be greatly appreciated.
• The Euler totient function, $$varphi$$, is a map from the positive integers to the positive integers. It is defined as follows: $$\varphi(n)$$ equals the number of numbers in $$\{1, 2, 3, \dots, n\}$$ that are relatively prime with $$n$$ (relatively prime means the only positive integer dividing it and $$n$$ is 1). For example, $$\varphi(12) = 4$$, as the only numbers in $$\{1, 2, \dots, 12\}$$ which are relatively prime to 12 are 1, 5, 7 and 11. Compute $$\varphi(n)$$ for a lot of values of $$n$$. Try to make some conjectures based on the structure of $$n$$. Can you prove any of them? We'll see this function when we talk about cryptography.

Description: There are many stages and approaches to solving a problem. Before trying to find a solution, it's often a good idea to determine if one exists, as if it doesn't our search will be fruitless! If we're fortunate enough to know there is a solution, the next step is to find it. Oftentimes we can solve a problem through brute force approaches, but these are inelegant, give us no greater understanding of the problem or why the solution works, and frequently cannot be generalized to related problems. The goal of this seminar is to introduce you to alternatives to brute force. There are many examples throughout mathematics and science where, if you look at the problem the right way, what was originally a long, involved calculation evaporates into a wonderful `A-ha!' moment! We'll draw examples from problems that can be solved by tedious, long computation as well as quick arguments if viewed the right way.

• Combinatorics:  Problems include:
• Formulas for the Fibonacci numbers (here's a nice video introduction): brute force recurrence versus divine inspiration to reach Binet's formula. Application includes double plus one (see this video for more on the problem).
• Products of binomial coefficients (see the wikipedia page for the definition). Examples include the formula for the sum of adjacent binomial coefficients, proof of the Binomial Theorem, as well as $$\sum_{k=0}^n \left({n \atop k}\right)^2$$ (with $$n$$ a positive integer). We can solve some of these by brute force, but storytime is much better.
• Two nice applications of the binomial theorem: if know calculus, it allows us to show $$e^x e^y = e^{x+y}$$; it's also used to compute the derivative of $$f(x) = x^n$$ for $$n$$ a positive integer.
• A nice application of the exponential relations is the proof of all trig identities; this requires $$e^{i \theta} = \cos\theta + i \sin\theta$$ (Euler's formula).
• Partition problems: the cookie problem or the stars and bars problem: how many ways are there to divide $$C$$ identical cookies among $$P$$ distinct people? As a nice application can analyze the probability of winning a lottery with and without repeated numbers allowed.
• Symmetry, Duality and Exhaustion: Problems include:
• Tic-tac-toe: how many first moves are there? How many games?
• Chess pieces: Consider a 5 x 5 chessboard; place 5 queens on the board so that 3 pawns may safely be put down.
• Legal 21: given the numbers 1, 5, 6 and 7, use each number once and only once and create 21 by combining these with the four basic operations (plus, minus, times, divide). Key is to enumerate all possibililties.
• Application: duality arises in linear programming (see this video for more), in signal processing, ....
• Dimensional analysis: Problems include:
• Prove the Pythagorean theorem: if $$a$$ and $$b$$ are the sides of a right triangle with hypotenuse $$c$$, then $$a^2 + b^2 = c^2$$.
• Show the period of a pendulum near the surface of the Earth is proportional to $$\sqrt{L/g}$$, where $$g$$ is the acceleration due to gravity and $$L$$ is the length of the pendulum.
• Morley's theorem: we probably won't do, but it's worth knowing; see this site for some proofs.
• Fast Algorithms: Problems include:
• Solving polynomial equations: we can search for integer solutions and try to factor by sight, but want a general formula. Quadratic equation is the solution. Start with simple linear and keep generalizing.
• Babylonian multiplication: leads to the concept of a look-up table. Sometimes some operations are cheaper than others, and want to exploit. Another example is the Strassen's algorithm for matrix multiplication.
• Evaluating a polynomial efficiently: Horner's algorithm. Application to fractal geometry and iterating polynomials. A great example is the Mandelbrot set (see this video); one of the first instances of fractals being used in movies was the Genesis torpedo in Star Trek II: The Wrath of Khan (click here for the video).
• Fast Exponentiation: one of the key ingredients in modern cryptography.
• The Euclidean Algorithm: allows us to find greatest common divisors, and runs in a lot less time than we might expect (every two iterations is a savings of a factor of 2).
• Let $$a$$ be relatively prime to a prime $$p$$. Can prove $$a^p - a$$ is a multiple of $$p$$. Without loss of generality may assume $$a$$ is in $$\{0, 1, \dots, p-1\}$$. We have $$a^p = (a-1 + 1)^p = (a-1)^p + p (\cdots) + 1^p$$ (this is because of the binomial theorem), so $$a^p = (a-1)^p + 1$$ plus a multiple of $$p$$. We do this a total of $$a$$ times and we get $$a^p = a$$ plus a multiple of p, proving the desired relation. After some obvious algebra, we get $$a \cdot (a^{p-1} - 1)$$ is a multiple of $$p$$, and since $$a$$ and $$p$$ are relatively prime, we find that $$a^{p-1} = 1$$ plus a multiple of $$p$$ if $$p$$ is a prime and $$a$$ is relatively prime to $$p$$. This gives a primality test (called the Fermat Primality Test). If we choose an integer $$n$$ and another integer $$a$$ relatively prime to $$n$$, then if $$a^{n-1}-1$$ is not a multiple of $$n$$ then $$n$$ cannot be prime. Sadly the reverse direction need not be true (see the Carmichael numbers).
• Note the above primality test can tell us an integer is prime without telling us a factor! This should seem strange: we can prove a factor exists without being able to find it! In general, factorization is believed to be hard, and is the foundation of many cryptographic schemes. We can talk about $$N = pq$$ for two large primes versus a large random number password.
• Key takeaway: there are faster ways of doing computations than you might expect!
• Probably won't do, but another topic would be the different sorting algorithms (BubbleSort, MergeSort, QuickSort, ...).
• Irrationality Proofs: Problems include:
• Standard proof that $$\sqrt{2}$$ is irrational. Give Tennenbaum's geometric proof and my generalization with David Montague (paper is here).
• Proofs by Induction: Problems include:
• Maybe sum of odd numbers by induction versus the geometric proof. Could also do triangular numbers by induction versus the geometric proof (use a right triangle and not an equilateral triangle and remove the double counting).
• Error Detecting / Correcting Codes: Problems include:
• Sphere packing, strange probabilities....

Math Riddles
I also maintain a math riddles page. Please feel free to share these riddles with your colleagues and your students, and let me know if there is anything I can do to make the site more useful for you and your classes. The goal is to continuing expanding the student / teacher's corner to facilitate using these in classrooms. If you or any of your students are interested in helping (a great item for the CV!), or want updates on the progress, let me know.