• Solving the Rubik's cube (thanks Alan Chang and Umang Varma!):
  • 2x2x2: https://www.youtube.com/watch?v=PKZ7pxFyYu0
  • 3x3x3: https://www.youtube.com/watch?v=FO1kOU-3Blw
  • 4x4x4: parity issue: https://www.youtube.com/watch?v=nC4jsBD1L1w
• Rubik's cube (and the G-d number), and homepage :https://www.rubiks.com/ 

• Gaussian integral: http://math.stackexchange.com/questions/34767/int-infty-infty-e-x2-dx-with-complex-analysis (see here for notes by Keith Conrad on more ways to do: http://www.math.uconn.edu/~kconrad/blurbs/analysis/gaussianintegral.pdf)

• Contour integration: https://en.wikipedia.org/wiki/Methods_of_contour_integration
• Key ideas: using differentiation to pull off the term we need in a series expansion, finding best way to complexify.

• The main idea in our proof of Sperner's lemma is a powerful one: a set whose cardinality is odd must be non-empty! This parity knowledge turns out to be exceptionally powerful. I know another situation where this is used: in the theory of elliptic curves one can create a function whose order of vanishing (conjecturally) encodes the order of the infinite part of the associated group of rational solutions to the elliptic curves. This has applications in terms of bounding the size of the class number. As these topics are a bit off the standard path for this course, I'll leave it at the following brief remark for now (we show a function \(f(x)\) is odd and that it has at least two zeros at \(x=0\), and thus it must have a third zero!); I am happy to write more; if you're interested just email me (either directly or through ephsmath@gmail.com) and I'll write back with additional details or post them here.
• Analysis review handout.
• One of the most important applications of contraction maps and fixed point theorems is to solving first order differential equations. There is a lot of great mathematics below the surface here.
  • Detailed notes here: http://web.williams.edu/Mathematics/sjmiller/public_html/209/handouts/firstorderexistunique.pdf (these go through the analysis preliminaries and the proof).
    • Lecture notes on the proof are pages 9-11 of this pdf file: http://web.williams.edu/Mathematics/sjmiller/public_html/209/209LectureNotes.pdf
    • Wikipedia page: http://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem
  • Example from previous year: I particularly liked seeing how quickly we converged to the solution to \(y'(t) = 2t(1 + y(t))\) with \(y(0) = 0\). We rewrote this as an integral equation: \(y(\tau) = \int_0^\tau 2t(1 + y(t))dt\), and notice that the solution to our differential equation is a fixed point to this integral equation! Here we have a map from functions to functions; this is a little different than what you've seen before, and serves as a nice introduction to functional analysis.
    • Integral equations: http://en.wikipedia.org/wiki/Integral_equation
    • Functional analysis: http://en.wikipedia.org/wiki/Functional_analysis
  • Video online here (from 2014): http://youtu.be/fso52yaDP78
• Applications of Sperner's Lemma:
  • Barycentric Coordinates, Fixed Point Theorem Preliminaries: Lecture from Operations Research Math 377:  https://youtu.be/gQiv7O6uLu0
    • Main items: https://en.wikipedia.org/wiki/Barycentric_coordinate_system
    • https://en.wikipedia.org/wiki/Simplex
    • We talked about conjugating maps to go from one space to another, this is a very important concept and allows us to reduce the analysis of arbitrary cases often to just one. See https://en.wikipedia.org/wiki/Topological_conjugacy for more. A particularly nice example of this is the Riemann Mapping Theorem.
    • We almost proved the Brouwer Fixed Point Theorem. We came close. I want to go slowly through the argument and highlight the ideas and why we approach the proof the way we do. This is a monumental result, and worth spending the time and doing it right, and seeing how one would choose to introduce the various concepts. The most important is the idea of the index of the mapping. It's similar to a lot of things we've seen throughout the semester. We're looking for the first occurrence of something that must happen. Notice how important it was that the sum of the components is always 1, and that they are non-negative. It suggests a similar result might fail if our space were....
  • Brouwer Fixed Point Theorem, Linear Programming Games: Lecture from Operations Research Math 377: https://youtu.be/lLDqCyt-aGc
    • We saw earlier why we needed at least continuity. Today we saw how the index worked, and how the Bolzano-Weierstrass theorem worked. Notice that this is a non-constructive proof, and that at each stage of the Spernerization we get a new triangle, not necessarily in the old.
      • Good catch that I was a bit careless and we are really doing Bolzano-Weierstrass THREE times with three sub-sequences....
    • We played, or discussed playing, IcoSoKu.
  • Application: Nash equilibrium: https://en.wikipedia.org/wiki/Nash_equilibrium (his thesis is here: scan: http://rbsc.princeton.edu/sites/default/files/Non-Cooperative_Games_Nash.pdf , slides on talk here: http://www.eecs.harvard.edu/~parkes/cs286r/spring02/papers/nash-cornell.pdf ).

Monday, April 10: Video: Complex Analysis in a day: The Path to Cauchy's Residue Formula: https://youtu.be/TGJtH7K-mXs

• Complex numbers: complex numbers are of the form \(z = x + iy\), with the complex conjugate \(z = x - iy\) with \(i^2 = -1\). While it is impossible to totally order the complex numbers (unlike the reals, which can), they still have many nice properties. They are associative and commutative. It is possible to generalize the complex numbers further. The first is the four dimensional space of the quaternions, which are associative but not commutative. The next generalization is the octonians, which are eight dimensional (over the reals) and now even associativity is lost! What's next? The sedenions!
• We showed the definition of the derivative can be rewritten and interpreted as the first order Taylor series does an excellent job approximating the function. Interestingly, a function can have continuous partial derivatives without being differentiable. The big theorem is that if the partial derivatives are continuous then the function is differentiable. It is possible for a function to have a limit along some paths but not others, or to have different limits along different paths. For a function to be differentiable, the quotient must tend to the same value no matter what path you take. In one variable there is essentially just two choices (approach from the left or the right); it's much more interesting in several variables, as Cam and Kayla's artwork show.
  • Here is a nice example: it has the same limit along any straight line or parabola, but a different limit along some cubic paths! Consider f(x,y) = (x⁸ + y⁸)/(x² + y⁸) - (x¹⁰ + y¹⁰)/(x⁴ + y¹⁰) and take the limit as (x,y) approaches (0,0).
• While the definition of complex differentiable looks innocuously similar to that for one variable, remember that the limit must hold for any path. This implies a variety of strange facts. A good way of viewing this is that functions of a complex variable \(z\) can be written as functions of \(x\) and \(y\). Using \(z = x + iy\) and \(\overline{z} = x - iy\), we have \(x = (z+\overline{x})/2\) and \(y = (z-\overline{z})/2i\). Essentially the complex differentiable functions are those which depend on \(z\) and not \(\overline{z}\). Thus only special combinations of x and y are allowed. Later we'll see that complex differentiability is equivalent to our function satisfying the Cauchy-Riemann differential equations. One way of interpreting this is that we have a function whose corresponding function of two real variables x and y has zero curl. This means we have restricted ourselves to a very nice subset of real valued functions, in particular ones where Green's Theorem (or Stokes' Theorem) is particularly easy to apply.
  • In comparing results in real analysis to complex analysis, we met some very interesting functions. One is sin(1/x); though this function isn't differentiable, x² sin(1/x) is (but not infinitely often!).
  • Another great example is the function f(x) = exp(-1/x²) for x not zero and 0 otherwise. To take the derivatives requires L'Hopital's rule. This function is extremely important in probability, as it shows that a Taylor series does not uniquely determine a function. In probability this arises as the moments of a probability distribution do not always uniquely determine the probability distribution. This is covered in great depth in some notes I wrote up for Math 341 (probability).
•  If you are unfamiliar with Green's theorem, watch my lecture on Green's Theorem in a day from Calc III (you can watch it before then if you'd like; this quickly goes over path integrals and explains the calculations we've done).
• One of the gems of complex analysis is Cauchy's Integral Theorem, A complex differentiable function satisfies what is called the Cauchy-Riemann equations, and these are essentially the combination of partial derivatives one sees in Green's theorem. In other words, the mathematics used for Green's theorem is crucial in understanding functions of a complex variable.
  • For me, I consider it one of the most beautiful gems in mathematics that we can in some sense move the derivative of the function we're integrating to act on the region of integration! This allows us to exchange a double integral for a single integral for Green's theorem (or a triple integral for a double integral in the divergence theorem). As we've seen constantly throughout the year, often one computation is easier than another, and thus many difficult area or volume integrals are reduced to simpler, lower dimensional integrals. This subject is done properly in differential forms.
• Recall the exponential function exp is defined by \(e^z = \exp(z) = \sum_{n = 0}^{\infty} z^n/n!\). This series converges for all \(z\). The notation suggests that \(e^z e^w = e^(z+w)\); this is true, but it needs to be proved. (What we have is an equality of three infinite sums; the proof uses the binomial theorem.) Using the Taylor series expansions for cosine and sine, we find e^(iθ) = cos θ + i sin θ. From this we find |e^(iθ)| = 1; in fact, we can use these ideas to prove all trigonometric identities! For example:
  • Inputs: e^(iθ) = cos θ + i sin θ and e^(iθ) e^(iφ) = e^(i (θ+φ))
  • Identity: from e^(iθ) e^(iφ) = e^(i (θ+φ)) we get, upon substituting in the first identity, that (cos θ + i sin θ) (cos φ + i sin φ) = cos(θ+φ) + i sin(θ+φ). Expanding the left hand side gives (cos θ cos φ - sin θ sin φ) + i (sin θ cos φ + cos θ sin φ) = cos(θ+φ) + i sin(θ+φ). Equating the real parts and the imaginary parts gives the identities
    • cos(θ+φ) = cos θ cos φ - sin θ sin φ
    • sin(θ+φ) = sin θ cos φ + cos θ sin φ
    • Here's a picture that shows the standard proof with a convoluted diagram.
  • One can prove other identities along these lines....

   

Friday, April 7: Video: Differentiating Identities and Probability: https://youtu.be/CU_UIS-4xFI

• Differentiating identities:
  • Here is a differentiating identies handout (this became the nucleus for the section in a book I wrote). If the sums are finite there is no difficulty justifying the method, but for infinite sums it is very important to check to make sure we can do this interchange (interchanging the sum and the derivative); it is frequently referred to as differentiating under the integral sign. Differentiating identities is a powerful technique; it creates infinitely many more identities from a given one.
  • The idea is that identities are hard to find, and if you can create more from one that's good! We started with the geometric series formula (with a hand-waving and then a rigorous proof), and talked about what we can get if we can interchange the infinite sum and the derivative. Later in class we showed that we can justify the interchange here as the tail of a geometric series is also a geometric series; this is a fortitous situation which we can exploit. In general one has to work harder.
    • Details of interchanging the sum and the derivative: https://www.youtube.com/watch?v=8dPhdo98Gk8&feature=youtu.be (Lecture from my Math 389 class).
  • We talked about some of the important standard distributions. We did the Geometric Distribution in class.
    • The first was the geometric distribution. If you thought the correct definition was to keep \(p\) the probability of success and have the density function \((1-p)^{n-1}p\), you're right! This is the accepted normalization. We keep \(p\) for sucess, ALWAYS. Also this way we talk about a success on toss \(n\), not on \(n+1\). Again, it doesn't matter which way you define it, but you want to be consistent with others.
    • Another is the exponential distribution. There are two `natural' definitions, depending on what we want the scale parameter to be. Personally, I prefer \((1/\lambda) \exp(-x/\lamba)\) to \(w exp(-xw)\) (this is one of the few times I disagree with Wikipedia); the first has mean \(\lambda\) while the second has mean \(1/w\). You always have to be careful when looking at different sources, as they could use different normalizations.
    • Binomial Distribution:
      • Wikipedia link: http://en.wikipedia.org/wiki/Binomial_distribution
      • Algebra: One of the hardest aspects of using the method of differentiating identities is to find the right identity and the simplest path through the algebra; fortunately it's also possible to reach the correct answer if you take a longer route! While we could have started with \(\sum_{k=0}\left({n \atop k}\right) p^k (1-p)^{n-k} = 1\), if we apply \(\frac{d}{dp}\) or \(p\frac{d}{dp}\) we have to use a product rule, and we'd like to avoid that. We can't just multiply through by \(k\) as that's our summation variable, and whatever we do to one side must be done to the other. A nice trick is to start with the Binomial Theorem:   \(\sum_{k=0}\left({n \atop k}\right) x^k y^{n-k} = (x+y)^n\), and now apply \(x\frac{d}{dx}\). The advantage is clear: no product rule! All we need to do is then set \(x = p\) and \(y = 1-p\). What's nice is that, even though at the end of the day we have two dependent terms (\(p, 1-p\)), we start with two independent quantities (\(x, y\)) for differentiation.
    • Next is the Poisson distribution. Wikipedia redeems itself and has a nice discussion of how it arises here. Didn't do this one in class, did it in 2015: https://youtu.be/YF0v4O9dwyc.
    • In looking at the two distributions above, our first though was to find the mean. We saw both were computable via differentiating identities. For the Poisson, we saw how to compute the mean (and even harder, the variance) by being clever about the algebra. For the mean we needed to notice that \(n / n!\) equals \(1/(n-1)!\). For the variance, the trick was to write \(n^2\) and \(n(n-1 + 1)\), and then when we divide by \(n!\) we get \(1/(n-2)! + 1/(n-1)!\), and we can handle each piece.
    • It's worth thinking about the variance of the Poisson. A major theme of the course is the need to be able to look at a lengthy equation and get a feel for what it's saying. The mean and the standard deviation are supposed to be in the same units, so if the mean is λ then shouldn't the standard deviation be λ, because if the variance were λ then the standard deviation would be λ^1/2 and that would have the wrong units, right? Wrong. For an exponential with density f(x) = λ exp(-λx) the mean and standard deviation are both 1/λ, and we can see that this is the correct λ dependence by scale issues: we exponentiate λx, so λx must be unitless so if x is in meters say then λ is in 1/meters, and thus this is the correct λ dependence for the mean and standard deviaton. What goes wrong for the Poisson? Remember the density there is f(n) = λⁿ e^λ / n!; here λ is alone in the exponential and is thus unitless! This means we can't use the unit analysis to say that the standard deviation and the mean have the same λ dependence.

    • The Poisson random variable often models the number of events in a window of time. Also, frequently normalized spacings between events converge to Poissonian (a great example is to look at the primes). Another is the spacings between the ordered fractional parts of \(n^k \alpha\) (click here for more).

  • General advice: to differentiate an identity,  you need an identity. Seems silly to state but it's essential. Often the hardest part of these problems is figuring out how to do the algebra in a clean way. For us, we saw that frequently we want to move the normalization constant over to the other side; it allows us to avoid a product or quotient rule. We also saw sometimes it's easier to computer \(E[X(X-1)]\) than \(E[X^2]\), and then do algebra. It all comes down to whether or not it's easier to apply \(d/dx\) or \(x d/dx\).

Wednesday, April 5: Video: Integration by Differentiating: https://youtu.be/yvfKdI9K-JU

• We evaluated the integral of \(\sin x / x\) by using the Laplace Transform, and then specializing to a specific value. This is a very common technique: we have a free parameter, find an answer as a function of that parameter, and then specialize to the value we need.
• Big theorems on interchanging limits and integrals:
  • Monotone convergence theorem: https://en.wikipedia.org/wiki/Monotone_convergence_theorem
  • Dominated convergence theorem: https://en.wikipedia.org/wiki/Dominated_convergence_theorem
  • Fatou's lemma: https://en.wikipedia.org/wiki/Fatou%27s_lemma
• Differentiating under the integral sign: https://en.wikipedia.org/wiki/Differentiating_under_the_integral_sign
• Differentiating identities and probability theory: See the Monday, October 17, 2016 additional comment from Probability: http://web.williams.edu/Mathematics/sjmiller/public_html/341Fa16/addcomments.htm
• Lecture proving the Fundamental Theorem of Calculus through the Mean Value Theorem: http://www.youtube.com/watch?v=Q1TQtH6POyI (52 minutes)
• Lecture proving Green's Theorem in a day: https://www.youtube.com/watch?v=Iq-Og1GAtOQ (51 minutes)

• General links:
  • Price Is Right Link: http://www.cnn.com/2013/11/18/showbiz/price-is-right-strategy/index.html?hpt=hp_t3
  • Williams vs Dimaggio Streak: http://www.thepostgame.com/features/201107/does-ted-williams-own-more-impressive-streak-joe-dimaggio
  • Consecutive times reaching first safely (without causing an out or by error): http://en.wikipedia.org/wiki/List_of_Major_League_Baseball_individual_streaks#Consecutive_plate_appearance_records
  • Interesting arXiv post -- the first author is a computer (his 'master' chooses where to put him based on whom he believes did the most work on the paper!): http://arxiv.org/pdf/1504.02513.pdf
• Today's lecture serves two purposes (slides here). Most importantly it introduces many of the key ideas and challenges of mathematical modeling. I give this lecture Calc III and Probability. Most students there won't be taking partial derivatives or integrals later in life (though you never know!); however, almost surely you'll have a need to model, to try and describe a complex phenomena in a tractable manner.
  • Sabermetrics is the `science' of applying math/stats reasoning to baseball. The formula I mentioned at the start of the semester is known as the log-5 method; a better formula is the Pythagorean Won - Loss formula (someone linked my paper deriving this from a reasonable model to the wikipedia page), the topic of today's lecture. ESPN, MLB.com and all sites like this use the Pythagorean win expectation in their expanded series. My derivation is a nice exercise in multivariable calculus and probability
  • In general, it is sadly the case that most functions do not have a simple closed form expression for their anti-derivative. Thus integration is magnitudes harder than differentiation. One of the most famous that cannot be integrated in closed form is \(\exp(-x^2)\), which is related to calculating areas under the normal (or bell or Gaussian) curve. We do at least have good series expansions to approximate it; see the entry on the erf (or error) function.
    • The anti-derivative of \(\ln(x)\) is \(x \ln(x) - x\); it is a nice exercise to compute the anti-derivative for \((\ln(x))^2\) for any integer \(n\). For example, if \(n=4\) we get \(24 x - 24 x \ln(x) + 12 x (\ln x)^2 - 4 x (\ln x)^3 + x (\ln x)^4\).
  • Another good distribution to study for sabermetrics would be a Beta Distribution. A nice example is the Laffer curve from economics. I would like to try to modify the Weibull analysis from today's lecture to Beta distributions. The resulting integrals are harder -- if you're interested please let me know.
  • Today we discussed modeling, in particular, the interplay between finding a model that captures the key features and one that is mathematically tractable. While we used a problem from baseball as an example, the general situation is frequently quite similar. Often one makes simplifying assumptions in a model that we know are wrong, but lead to doable math (for us, it was using continuous probability distributions in general, and in particular the three parameter Weibull). For more on these and related models, my baseball paper is available here; another interesting read might be my marketing paper for the movie industry (which is a nice mix of modeling and linear programming, which is the linear algebra generalization of Lagrange multipliers).
    • One of the most important applications of finding areas under curves is in probability, where we may interpret these areas as the probability that certain events happen. Key concepts are:
      • Probability distribution
      • Mean or Expected Value
      • Standard Deviation
      • Independence
      • Skewness and kurtosis (for the hypercompetitive students who really want to compare themselves to the class)
    • The more distributions you know, the better chance you have of finding one that models your system of interest. Weibulls are frequently used in survival analysis. The exponential distribution occurs in waiting times in lines as well as prime numbers.
    • In seeing whether or not data supports a theoretical contention, one needs a way to check and see how good of a fit we have. Chi-square tests are one of many methods.
    • Much of the theory of probability was derived from people interested in games of chance and gambling. Remember that when the house sets the odds, the goal is to try and get half the money bet on one team and half the money on the other. Not surprisingly, certain organizations are very interested in these computations. Click here for some of the details on the Bulger case (the bookie I mentioned in class is Chico Krantz, and is referenced briefly).
    • Any lecture on multivariable calculus and probabilities would be remiss if it did not mention how unlikely it is to be able to derive closed form expressions; this is why we will study Monte Carlo integration later. For example, the normal distribution is one of the most important in probability, but there is no nice anti-derivative. We must resort to series expansions; that expansion is so important it is given a name: the error function.
    • I strongly urge you to read the pages where we evaluate the integrals in closed form. The methods to get these closed form expressions occur frequently in applications. I particularly love seeing relations such as \(1/c = 1/a + 1/b\); you may have seen this in resistors in parallel or perhaps the reduced mass from the two body problem (masses under gravity). Extra credit to anyone who can give me another example of quantities with a relation such as this.
    • Click here for a clip of Plinko on the Price I$ Right, or here for a showcase showdown.
  • We discussed how website like ESPN and MLB have a very limited space to display information, especially if it's for a smart phone. Thus one cannot show every statistic, and we have to pick and choose which ones are worth showing. In one section I made a joke about including the team names, but this is actually a serious comment! The MBTA (or MTA for us old folk!) had a contest on how to redesign their subway map of Boston. Below are links to an interesting article on the subject and the maps.
    • http://www.theatlanticcities.com/commute/2013/04/dear-boston-heres-better-transit-map-you/5260/
    • http://www.theatlanticcities.com/commute/2012/01/translating-american-highway-system-subway-map/995/
    • http://www.cambooth.net/archives/850

Friday, March 17: Video: Integration Techniques: https://youtu.be/MH3tDaarOas

• We discussed how to use symmetry to evaluate some integrals.
  • Some nice examples: http://www.math.illinois.edu/~ajh/putnam/training/integrals1.pdf
  • More problems: http://www.math.nyu.edu/faculty/hausner/Putnamintegrals.pdf  and http://www.math.uchicago.edu/~fcale/Putnam/integrals.pdf
  • List of Putnam problems and what kinds of integration work: http://www.math.cornell.edu/~putnam/int_sum.html
• For a terrific application of symmetry or splitting up integrals, see the proof of the functional equation of the Riemann zeta function via theta functions: https://en.wikipedia.org/wiki/Riemann_zeta_function#Theta_functions
• We also discussed what I'm calling the `Bring it Over' method, where one has the same expression on both sides but with different weights. Note this is similar to solving algebra equations such as \(3x + 7 = 8 x - 9\).

Wednesday, March 15: Video: Maximizing Product, Babylonian Math, Jug Problem: https://youtu.be/VEdhSgdf82c

• Looked at the problem maximizing a restricted product.

  • We studied one of my favorite problems, given \(S = a_1 + \cdots + a_n\) with each \(a_i\) a positive integer, maximize the product of the \(a_i\). We quickly see the optimal is when each \(a_i\) is 2 or 3, and since \(2\ast 2 \ast 2 < 3 \ast 3\) we want \(3\)'s over \(2\)'s. We converted to a real problem and assumed there were \(n\) summands, each a real number. We got a function defined on the integers to maximize, replaced it with a function defined on the reals so calculus would be applicable. We then curve sketched and saw the function was increasing to its maximum and decreasing past it, so the optimal integer soln was either to the left or right of the optimal real soln (here optimal soln is referring to the number of summands). It's unusual to be this fortunate.

  • We had to maximize \(a_1 \ast \cdots \ast a_n\) given \(a_1 + \cdots + a_n = S\) and each \(a_i > 0\). We can do this with Lagrange multipliers, or since each \(a_i\) is in \([1, S]\) we can appeal to the \(n=2\) case because a real continuous function on a compact set attains its max and min. What is nice is that this existence result from real analysis improves to being constructive; if we were at the optimal point and all coordinates were not equal, we could simply replace two of them with the average and improve the product.

  • A nice application of this problem is that for disk storage (see radix economy), base 3 has advantages over base 2, though base 2 has the very fast binary search. Another nice example of base 3 occurs with the Cantor set.

• Did Babylonian Mathematics to follow our discussion on bases. We talked about how the Babylonians did mathematics. In addition to the horrors of base 60 (which Wikipedia tells me is due to the Sumerians, which must be true if they've posted it), they did give us the look-up table. The point is to reduce long, painful calculations to pre-computed quantities, with perhaps some (hopefully linear) interpolation as you can't pre-compute everything. In base 60, one would need to have tables for about 3600/2 = 1800 multiplications to be able to do \(xy\); the Babylonians noticed \(xy = ((x+y)^2 - x^2 - y^2) / 2\), or even better \(xy = ((x+y)^2 - (x-y)^2)/4\); this reduces the problem to knowing just squares (only 60 entries needed) and the ability to subtract and divide by 2. It's much better to do these simpler problems than the original harder one.

• We ended with a nice presentation on the jug problem, and the differences between DFS (depth first search) and BFS (breadth first search).

  • No discussion of the jug problem would be complete without this clip from Die Hard III: https://www.youtube.com/watch?v=5_MoNu9Mkm4

Monday, March 13: Video: Conway Checker Problem, \(e^\pi\) vs \(\pi^e\): https://youtu.be/PcK3HKXc4-A

• Conway's checker problem:
  • https://en.wikipedia.org/wiki/Conway's_Soldiers (see also links there)
  • http://www.cut-the-knot.org/proofs/checker.shtml
  • Good problem: Generalize to 3dimensions, Generalize to "double" jump: x x x [empty] -> [empty] [empty] [empty] x
• \(e^\pi\) vs \(\pi^e\).

  • We talked a lot about how to use logarithms to make an analysis easier, or to exponentiate. For example, \((S/x)^x = \exp(x \log(S/x))\).

  • Here is a warning that you may have been fooled into believing you learned certain derivatives when you hadn't. For example:

    • \(f(x) = x^n\) has derivative \(n x^{n-1}\). This follows from the definition of the derivative and the binomial theorem to expand \((x+h)^n\) when \(n\) is a positive integer.

    • \(f(x) = x^{p/q}\) has derivative \(\frac{p}{q} x^{p/q-1}\). This follows by setting \(g(x) = f(x)^q = x^p\) and then differentiating, which gives \(g'(x) = q f(x)^{q-1} f'(x) = p x^{p-1}\), and then substituting and solving for \(f'(x)\). We cannot get it the same was as the derivative of \(x^n\), as that would require knowing the binomial theorem for non-integral exponents.

    • \(f(x) = x^{\sqrt{2}}\) has derivative \(\sqrt{2} x^{\sqrt{2}-1}\). This follows from using the exponential function and the chain rule: \(x^{\sqrt{2}} = \exp(\sqrt{2} \log x)\).

    • Thus, \(x^r\) does have derivative \(r x^{r-1}\), but the proof for general \(r\) goes through the exponential function.

  • We also used logarithms in the elementary approach to \(e^\pi\) vs \(\pi^e\). Here is a nice argument, using \(e > \pi - e > 0\).
    • \(e^\pi\) vs \(\pi^e\): take logarithms of both sides.
    • \(\pi \log e = \pi\) vs \(e \log \pi = e \log(e + \pi - e)\), where the last follows form adding zero. We do this so we can exploit log laws (log of a product is sum of the logs).
    • \(\pi\) vs \(e \log e + e \log\left(1 + \frac{\pi - e}{e}\right)\), and now we use \(\log(1+x) = x - x^2/2 + x^3/3 - x^4/4 = x\) plus a negative number (decreasing in absolute value, alternating).
    • \(\pi\) vs \(e + e \frac{\pi -e}{e}\) + negative number.
    • \(\pi\) vs \(\pi\) + negative number; thus it must be \(\pi \ge \pi\) + negative number, and thus working upward we find \(e^\pi \ge \pi^e\).
  • Another approach to that problem was to look at \(f(x) = e^x - x^e\) and \(g(x) = e^{x/e} - x\). I find the second function easier to work with. Note \(g(0) = 1 > 0\) and \(g(e) = 0\). We can figure out what's happening by looking at the derivative: \(g'(x) = e^{-1} e^{x/e} - 1\). Note \(g'(x) < 0\) for \(0 \le x < e\) and \(g'(x) > 0\) for \(x > e\). Thus \(g(x)\) is monotonically decreasing to \(x=e\), where it is zero, and then increases. Note the first derivative allows us to get a good feel for the shape of the function, and thus all other \(x\) have \(g(x) > 0\), so \(e^{\pi/e} > \pi\), which after exponentiation gives \(e^\pi > \pi^e\).

Friday, March 10: Video:  Monovariants: Tournament, Cards, Conway's Checker Problem: https://youtu.be/iKqhjKmI5x4

• We solved the card sort problem by induction. It can also be done with a monovariant approach. Recall the problem: Take the numbers 1, 2, ..., \(n\) and randomly sort them. If the number \(k\) is on top, reverse the order of the first \(k\) numbers. Continue as long as the top number isn't a 1. Prove that eventually this process terminates with a 1 at the top. (So, if we start with the ordering 4 2 1 3 5, the next ordering is 3 1 2 4 5, then 2 1 3 4 5 and finally 1 2 3 4 5).
  • I think the following solution works. It clearly suffices to show that eventually either \(1\) or \(n\) is at the top, or \(n\) is at the bottom (if it's 1 we win, while if \(n\) is at the top then on the next turn it's on the bottom, and an \(n\) on the bottom reduces us to the case of \(n-1\) and we are done by induction).
    • I claim that either 1 is eventually on top, or \(n\) is eventually on the bottom. Let us assume this does not happen for some \(n\), and without loss of generality let \(N\) be the smallest integer such that \(1\) is never on top and \(N\) is never on the bottom. As there are only finitely many ways to order \(N-1\) numbers, the ordering must be cycle (and in all the elements of the cycle, \(N\) is never on the top or bottom).
    • We have \(N\) is never on the top or the bottom; thus whatever card is on the bottom must always stay on the bottom; let's call that card \(k\). Since it is only the top card that causes change to happen, note that we could safely switch \(N\) and \(k\). If we do this, we would still have an infinite cycle but now it would only involve the numbers 1, 2, ..., \(N-1\). This contradicts the minimality of \(N\) and thus our assumption is wrong.
  • Suggestion from class (John): About the monoinvariant: I think if you compile a list of all the elements that are in order (i.e. 5 is in the 5th place) and order them from the largest to the smallest, this list's dictionary order (i.e. compare the first element, if they are the same compare the second) is always increasing.
• Conway's checker problem: SPOILER ALERT!
  • https://en.wikipedia.org/wiki/Conway's_Soldiers (see also links there)
  • http://www.cut-the-knot.org/proofs/checker.shtml

Wednesday, March 8: Video: Chessboard Tiling, Room Movement, Reversing Cards: https://youtu.be/QYNBKXFfVmA

• Monoinvariants are a wonderful topic, one often not seen (sadly) in the curriculum. We talked about how it allows us to solve the virus propagation problem, but this is just the beginning.
  • See Noether's Theorem for applications of conserved quantities in physics: http://en.wikipedia.org/wiki/Noether's_theorem
  • Slides on talk on Invariants, Monoinvariants and Extrema: http://www.mit.edu/~pengshi/math149/talk2.pdf
  • Berkeley Math Circle Lecture Notes on Monoinvariants: http://mathcircle.berkeley.edu/BMC3/monovar.pdf
  • Problems on invariants and monoinvariants: http://math.stanford.edu/~vakil/putnam07/07putnam5.pdf
• We started with a chessboard problem and creating a monoinvariant to show we cannot cover the board with 2 opposite squares missing with 1x2 dominoes. We could also have called this a parity argument; we'll see more of such arguments as the semester continues.
• We proved a nice theorem: if \(x_i \ge 0\) then the maximum value of \(x_1^2 + \cdots + x_r^2\) subject to \(x_1 + \cdots + x_r = N\) is when one is \(N\) and the rest vanish. One proof was to note without loss of generality \(N=1\) and then see that if \(0 < x_1, x_2 < 1\) then \(x_1^ + x_2^2 < x_1 + x_2\) so the sum of the squares is strictly less than 1. We could also use the monovariant argument from class. We have to be a bit careful, as we are often using a great result from real analysis: a continuous function on a closed, bounded set attains its maximum and minimum (to see why we need the interval to be closed, consider \(f(x) = (x-1/2) / (x(1-x)\). As \(x \to 1\) we have \(f(x)\) approaches positive infinity, while if \(x \to 0\) we approach negative infinity. For more on this see the Extreme Value Theorem.
• More chessboard problems: http://www.cut-the-knot.org/do_you_know/chessboard.shtml

Friday, March 3: Video: 12a: Monovariants, Zeckendorf Decomposition: https://youtu.be/kC7GVWEaVYI (slides HERE, start at 16minutes, then go to start); 12b: Pigeon Hole Problem, Dirichlet's Approximation Theorem: https://youtu.be/AA0ajr-foUw

• Readings on monovariants:
  • Theory and examples: http://www.mit.edu/~pengshi/math149/talk2.pdf
  • Examples and problems: http://www.math.kent.edu/~soprunova/64091s15/monovariants15.pdf
  • Problems: http://math.stanford.edu/~vakil/putnam07/07putnam5.pdf
• Dirichlet's approximation theorem: https://en.wikipedia.org/wiki/Dirichlet's_approximation_theorem (if \(\alpha\) irrational then there exists integers such that \(|\alpha - p/q| < 1/q^2\).
• Hurwitz' theorem (improves bound to \(1 / \sqrt{5} q^2\):  https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(number_theory)
• P:roject Euler problems: https://projecteuler.net/archives

Wednesday, March 1: Video: Irrationality Proofs, Morley's Theorem, Pigeon Hole Problems: https://youtu.be/xx8f8z8zzfQ

• Rational irrational proofs (with David Montague), Mathematics Magazine (85 (2012), no. 2, 110--114).  pdf (expanded version: pdf;    letter from Nerode and Tennenbaum on history of proof: pdf)
  • See also http://www.cut-the-knot.org/proofs/GraphicalSqRoots.shtml
• Irrationality of sqrt(2):
  • standard proof: https://www.math.utah.edu/~pa/math/q1.html
  • geometric proof: https://jeremykun.com/2011/08/14/the-square-root-of-2-is-irrational-geometric-proof/
• Morley's Theorem: John Conway's simple proof is here (just search the pdf for Morley; interestingly it also has the irrationality of sqrt(2)).
• Pigeon hole principle: Lots of great links for readings and problems.
  • http://math.stackexchange.com/questions/194312/show-me-some-pigeonhole-problems
  • http://www.math.lsa.umich.edu/~hderksen/ProblemSolving/PS7.pdf
  • http://artofproblemsolving.com/wiki/index.php/Pigeonhole_Principle
  • http://mindyourdecisions.com/blog/2008/11/25/16-fun-applications-of-the-pigeonhole-principle/
  • http://www.math.ucla.edu/~radko/circles/lib/data/Handout-123-153.pdf

Monday, February 27: Video: Geometry Problems, Points on Spheres and Hemispheres: https://youtu.be/8jRZjc9Q8PU

• Today one of the key, new ideas was the pigeonhole principle, or Dirichlet's Box principle. There are two main steps: identify the pigeons, identify the boxes. Usually the pigeons are obvious, but it can be very hard to find the boxes.
  • The original version says if we have N+1 pigeons sent to N boxes, at least one box gets 2 pigeons. More generally, if M > N and M pigeons are sent to N boxes, at least one box gets ceiling(M/N) pigeons (ceiling(x) means the smallest integer at least as large as x). This version is often very useful.
  • Wikipedia entry: http://en.wikipedia.org/wiki/Pigeonhole_principle
  • Fun reading:
    • http://www.cut-the-knot.org/do_you_know/pigeon.shtml
    • http://www.math.ust.hk/~mabfchen/Math391I/Pigeonhole.pdf (more extensive notes)
    • http://zimmer.csufresno.edu/~larryc/proofs/proofs.pigeonhole.html (a friendly version)
  • Video online here from the 2014 iteration of the class: Pigeonhole Principle: Part 1: http://youtu.be/b_7Gam3o_eA      Pigeonhole Principle: Part 2: http://youtu.be/jHomcvVEf2w
• We may have proved a new theorem; I wouldn't be surprised if it's been done before, but I think it's worth writing up as a problem and / or short note, say for the Pi Mu Epsilon journal: given a positive integer m, if there are 2m-2 points on a unit circle then at least m of them are on the same closed semi-circle, and it is possible no m satisfy such a relation if there are 2m-3 points. Anyone want to take point in writing this up? I saw many websites with the original problem but not with generalizatoins: http://math.stackexchange.com/questions/675646/if-there-are-5-points-on-a-sphere-then-4-of-them-belong-to-a-half-sphere (we generalized in 2-dimensions, how hard would it be in higher)? Working on this can replace doing standard HW problems.

Friday, February 24: Video: Pythagorean Formula, Asking Questions: https://youtu.be/6HxVSoOR-zc (click here for slides)

• Pythagorean Theorem
• Fermat's Last Theorem
• Beal's conjecture
• spherical coordinates, coordinates, area and volume of the n-sphere
• hyperbolic functions, hyperbolic space

Wednesday, February 22: Video: Catalan numbers, Generating Functions, Circle Method: https://youtu.be/KfEig_0kckQ

• We talked about the generating function approach to Waring's problem, which is quite challenging. Interestingly, a slightly different version of that problem (not surprisingly known as the Easier Waring's Problem), is elementary and easy: https://uniformlyatrandom.wordpress.com/2008/11/14/the-easier-warings-problem/
• The Circle Method allows us to attack a variety of problems.
  • Rubinstein paper on the Hardy-Littlewood Constant and Twin Primes:  http://www.jstor.org/stable/2324298
    • Direct link to Rubinstein's paper (not sure if this will work): http://www.jstor.org/stable/pdfplus/2324298.pdf?acceptTC=true&jpdConfirm=true
    • Note from one of my students: http://web.math.princeton.edu/mathlab/projects/primes/ds/ds_primes.doc
  • Rubinstein and Sarnak: Chebyshev's bias: http://www.math.uwaterloo.ca/~mrubinst/publications/Chebyshev.pdf
  • Hardy-Littlewood Conjectures:
    • Twin prime conjecture: http://en.wikipedia.org/wiki/Twin_prime#First_Hardy.E2.80.93Littlewood_conjecture
    • k-tuple conjecture: http://mathworld.wolfram.com/k-TupleConjecture.html (notice the integral!).
    • Prime constellation: http://mathworld.wolfram.com/PrimeConstellation.html
      • For Star Trek fans: USS Constellation: NCC 1017: http://mathworld.wolfram.com/PrimeConstellation.html
        • From MemoryAlpha: The Constellation studio model was constructed from a 1966 first edition AMT Enterprise model kit , no. S921. (Star Trek Encyclopedia, 3rd. ed., p.
          85) That particular edition sported a decal sheet with only the "NCC-1701" decal, and, with very limited options, its numbers had to be rearranged to create the unusually low "NCC-1017" Constellation registry number, the first new registry actually seen on a ship and, as it turned out, also the only time in the original airing of the Original Series , though 
          considered somewhat incongruous by many, due to the perceived discrepancy in the numbering system.

Monday, February 20: Video: Generating Functions, Binet's Formula, Waring's Problem, Catalan Numbers: https://youtu.be/nFDLwoTanDc

• Generating functions: There are many ways to prove Binet's formula for an explicit, closed form expression for the n-th Fibonacci number. One is through divine inspiration, the second through generating functions and partial fractions. Generating functions occur in a variety of problems; there are many applications near and dear to me in number theory (such as attacking the Goldbach or Twin Prime Problem via the Circle Method). The great utility of Binet's formula is we can jump to any Fibonacci number without having to compute all the intermediate ones. Even though it might be hard to work with such large numbers, we can jump to the trillionth (and if we take logarithms then we can specify it quite well).
  • Generating Functions Handout: This is from a book I'm writing on probability. The first section is motivation, feel free to skim. Section 19.2 is the most important. Section 19.3  is more technical and included for completeness; we won't cover. Section 19.4 talks about convolutions -- we'll need the very beginning to analyze the Catalan numbers.
• Catalan numbers: A nice generalization of these occur in a paper I wrote with some students and my kids.
• Waring's problem: We saw how to use generating functions to solve Waring's problem when \(k=1\); remember Waring's problem is whether or not for a fixed \(s, k\) there is a solution in non-negative integers to \(x_1^k + \cdots + x_s^k = n\) for either all \(n\) or perhaps all \(n\) sufficiently large. When \(k=1\) this is the Cookie Problem, and we could also solve combinatorially. If \(k > 1\) we no longer have a combinatorial approach, but generating functions do help.
• For more on generating functions you can see some videos of mine on Zeckendorf decompositions:
  • Cookie Monster Meets the Fibonacci Numbers. Mmmmmm -- Theorems!: http://youtu.be/5e6HsfxqVSE (slides here).

Wednesday, February 15. Video: Applications of the AM-GM Inequality, Zeckendorf and kilometers: https://youtu.be/MxZIk-Y0_U4

• More on Arithmetic Mean - Geometric Mean inequality. This is one of the most important inequalities in mathematics, and is a major tool in many contest problems.
  • Nice page on the AM-GM and its use in problem solving: http://jwilson.coe.uga.edu/EMT725/AMGM/SSMA.Bnghm.html
  • Wikipedia page: http://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means
  • My notes: http://web.williams.edu/Mathematics/sjmiller/public_html/OSUClasses/487/ArithMeanGeoMean.pdf
  • Some applications in math contests: http://www.austms.org.au/Gazette/2006/Mar06/boskoffS.pdf
  • Some more straightforward examples: https://brilliant.org/discussions/thread/arithmetic-mean-geometric-mean-am-gm/
  • Heron's formula: http://en.wikipedia.org/wiki/Heron's_formula
  • Olympiad inequalities: http://www.aam.org.in/site/st_material/14.pdf
  • Generalization of AM-GM: I. Ben-Ari and K. Conrad: https://arxiv.org/pdf/1402.0208.pdf
• Big item in the problems we studied is symmetry, especially if not symmetric how to symmetrize.
• Key ingredient is Heron's formula: http://jwilson.coe.uga.edu/EMT725/Heron/HeronProofAlg.html
• Isoperimetric inequality

Monday, February 13. Video: AM-GM inequality, Games (Triangle, Rectangle, SpotIt): https://youtu.be/fW28E5ruG3E

• Did more on AM-GM; see links from Friday, Feb 10
, especially my notes: http://web.williams.edu/Mathematics/sjmiller/public_html/OSUClasses/487/ArithMeanGeoMean.pdf.
• When we did the proof by Lagrange multipliers, it was important to know that there was a max/min. The fact that there are such points is the Extreme Value Theorem. Technically the conditions are not met for us unless we allow \(a_i = 0\), but we can always do by taking the \(a_i \ge \epsilon\) for some very small and positive \(\epsilon\).
• We proved it using 'hopping' induction. Lots of types of induction. I think this is cool: do powers of 2 and then fall back. Another interesting technique is transfinite induction.
• Fibonacci spiral:    https://www.youtube.com/watch?v=kkGeOWYOFoA
• Mathematics of SpotIt: http://math.stackexchange.com/questions/36798/what-is-the-math-behind-the-game-spot-it (spoiler alert -- solution here).
• The triangle game is related to Sperner's lemma and the Brouwer fixed point theorem....
• We didn't get to it, but the last item would have been fun versions of tic-tac-toe. We'll hit these later, but if you're interested here's some reading.

  • Earlier we talked about tic-tac-toe today as a counting problem: how many `distinct' games are there. We are willing to consider games that are the same under rotation or reflection as the same game; see http://www.btinternet.com/~se16/hgb/tictactoe.htm for a nice analysis, or see the image here for optimal strategy.

    • Probably the most famous movie occurrence of tic-tac-toe is from Wargames; the clip is here (the entire movie is online here, start around 1:44:17; this was a classic movie from my childhood).

    • Develin and Payne: bidding tic-tac-toe analysis

    • Gobble tic-tac-toe: https://www.fatbraintoys.com/toy_companies/blue_orange/gobblet_gobblers.cfm

Friday, February 10. Video: Recurrences and Arithmetic Mean - Geometric Mean inequality: https://youtu.be/caL16-pST5A 

• The first main item today was the Fibonacci series (an example of a difference or recurrence equation).
  • Recurrence relations: http://en.wikipedia.org/wiki/Recurrence_relation
  • Recurrence relations with multiple indices: http://www.researchgate.net/profile/Toufik_Mansour/publication/230802660_Recurrence_relations_with_two_indices_and_Even_trees/links/02bfe5110ea59d3674000000.pdf
  • Binet's formula: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibFormula.html
  • Application of Fibonacci numbers in Roulette! See https://www.youtube.com/watch?v=Esa2TYwDmwA 
• The second main item we studied today was the Arithmetic Mean - Geometric Mean inequality. This is one of the most important inequalities in mathematics, and is a major tool in many contest problems.
  • Nice page on the AM-GM and its use in problem solving: http://jwilson.coe.uga.edu/EMT725/AMGM/SSMA.Bnghm.html
  • Wikipedia page: http://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means
  • My notes: http://web.williams.edu/Mathematics/sjmiller/public_html/OSUClasses/487/ArithMeanGeoMean.pdf
  • Some applications in math contests: http://www.austms.org.au/Gazette/2006/Mar06/boskoffS.pdf
  • Some more straightforward examples: https://brilliant.org/discussions/thread/arithmetic-mean-geometric-mean-am-gm/
  • Heron's formula: http://en.wikipedia.org/wiki/Heron's_formula
  • Olympiad inequalities: http://www.aam.org.in/site/st_material/14.pdf
• Generalization of AM-GM: I. Ben-Ari and K. Conrad: https://arxiv.org/pdf/1402.0208.pdf

• Purpose of today was to do basic coding. Below are lectures from various classes
  • 2017 Problem Solving (331) / Number Theory (313): Mathematica code   (the pdf)      Video: 313: https://youtu.be/khMJuFXOhFE   Video 331: https://youtu.be/vPGjly_7QVw
  • 2016 Probability Iteration: Click here for the Mathematica code                Click here for a pdf        Video: https://youtu.be/JMhkq_bczKk
  • 2015 Probability Iteration: Click here for the Mathematica code                Click here for a pdf        Video:  https://youtu.be/wUC-iUsZL6M
  • some suggested problems you should try (click here for Mathematica notebook, and click here for a pdf).
• We discussed the cookie problem, counting the number of ways to divide 10 identical cookies among 5 distinct people (here's the classic clip where Cookie Monster meets the Count, whose full name is Count von Count -- they changed how he appears!). Counting is very important; we discussed how we could (and SHOULD) do simpler variants to build intuition. It's usually called the stars and bars problem. What I love here is the power of changing your perspective -- we go from a very painful brute force approach to being able to solve it in one line.
  • I have posted the cookie problem on my math riddles page (email me if you want to contribute); someone with far more patience than I solved it by brute force. Here's their solution. The final number, tabbed from the others, are how many distinct rearrangements we have of this basic configuration. The total  is 1001, or (10+5-1 choose 5-1). We saw this problem lead to a discussion of multinomial coefficients. When there are two people getting cookies, say 8 and 2, there aren't (5 choose 2) ways to assign people, but (5 choose 2) * 2! (we choose the 2 people, then there are 2! ways to choose which gets the 8 and which gets the 2). If we have 8 1 1 it's more involved. In that case it's (5 choose 3) to choose the three people, 3! ways to order which of the people gets which number, but then we must divide by 2! (as the two people getting 1 are indistinguishable). A better way to view 3!/2! is 3! / (2! 1!) (note the numbers on the bottom sum to the top). This is an example of a multinomial coefficient, a generalization of binomial coefficients. For example, if we have MISSISSIPPI, there would be 11! ways to order the letters (order matters) if the letters are distinguishable, but they're not. So let's put subscripts on the letters: MI₁S₁S₂I₂S₃S₄I₃P₁P₂I₄. We then have 4! ways of placing the four marked S's in the four S positions, and so on, giving 11! / (4! 4! 2! 1!) (I like including the final 1 so that the bottom sums to the top).
  • Below are the person's solutions, arranged differently than we did in class.
  • 0 0 0 0 10         5
    
    0 0 0 1 9         20
    0 0 0 2 8         20
    0 0 0 3 7         20
    0 0 0 4 6         20
    0 0 0 5 5         10
    
    0 0 1 1 8         30
    0 0 1 2 7         60
    0 0 1 3 6         60
    0 0 1 4 5         60
    0 0 2 2 6         30
    0 0 2 3 5         60
    0 0 2 4 4         30
    0 0 3 3 4         30
    
    0 1 1 1 7         20
    0 1 1 2 6         60
    0 1 1 3 5         60
    0 1 1 4 4         30
    0 1 2 2 5         60
    0 1 2 3 4         120
    0 1 3 3 3         20
    0 2 2 2 4         20
    0 2 2 3 3         30
    
    1 1 1 1 6         5   
    1 1 1 2 5         20
    1 1 1 3 4         20
    1 1 2 2 4         30
    1 1 2 3 3         30
    1 2 2 2 3         20
    
    2 2 2 2 2          1
  • What we're really doing is solving the equation \(x_1 + \cdots + x_5 = 10\) in non-negative integers. This is a very special type of Diophantine equation. It's actually a special case of Waring's problem, which looks at solving \(x_1^k + \cdots + x_s^k = n\) for fixed \(s\) and \(k\). These problems are in general not accessible through combinatorics; the case \(k=1\) is special. The general approach proceeds via generating functions, which we will cover in great detail later in the semester (it's one of the key concepts of the class).
  • Our solution to the cookie problem is quite elegant, and in some respects reminiscent of geometry class (remember all those proofs where the teacher cleverly adds auxiliary lines; the difference here is we just add more cookies). While it is possible to solve many combinatorial problems by brute force in principle, in practice this is not a good way to go -- it is time consuming, and quite likely that one makes a mistake. Typically one finds a way to interpret a given quantity two ways; we can compute one of them and thus we obtain a formula for the other. For example, we showed the number of ways of dividing C cookies among P people is (C + P - 1 choose P-1); here all the identical cookies are divided. What if we don't assume all the cookies are divided -- what is the answer now? It is just Sum_{c = 0 to C} (c + P - 1 choose P - 1); this is because we are just going through all the cases (we give out no cookies, 1 cookie, ...). What does this sum equal? Imagine now we have another person, say the Cookie Monster (this is one of Cameron's favorite clips), who gets all the remaining cookies. Then dividing at most C cookies among P people is the same as dividing exactly C cookies among P+1 people, and hence our sum equals (C + P+1 - 1 choose P+1 - 1).
  • Related to the cookie problem is the partition problem: for the cookie problem we consider 2+3+3+1+1 different from 1+2+3+3+1 as the five people are distinct; if we don't consider these distinct then we have a partition problem. It's a lot more complicated to count these, but some great mathematics (such as Young tableau).

• We talked about how to analyze expressions such as \((a+b)(b+c)(c+a) \ge 8abc\) for \(a, b, c > 0\). Moments like this are the reason for this class, talking about how to view. There's symmetry. Dimensional analysis shows we can rescale and assume either \(abc = 1\) or \(a+b+c = 1\); turns out the second is more useful and sets the stage to investigate it with Lagrange Multipliers.

• A common image for Mathematical Induction is that of following dominoes.

• We did a bit on Power Sums. Lot of great math behind finding the formulas, though if we assume it's a polynomial we can guess the coefficients by checking a few values, and then prove by induction. Note the constant term should be zero and calculus suggests the leading term of the sum of \(n^{p-1}\) for \(n = 1\) to \(N\) should be \(N^p/p\).

• Online Encyclopedia of Integer Sequences (homepage is http://oeis.org/ ) is a tremendous resource. You can enter the first few terms of an integer sequence, and it will list whatever sequences it knows that start this way, provide history, generating functions, connections to parts of mathematics, .... This is a GREAT website to know if you want to continue in mathematics. There have been several times I've computed the first few terms of a problem, looked up what the future terms could be (and thus had a formula to start the induction).

• Lots of good webpages with induction problems:

  • http://www.math.ucla.edu/~radko/circles/lib/data/Handout-142-159.pdf

  • http://math.stackexchange.com/questions/294677/looking-for-induction-problems-that-are-not-formula-based

  • http://www.math.cornell.edu/~mec/2008-2009/ABjorndahl/ppmi.pdf

  • http://staffhome.ecm.uwa.edu.au/~00021149/Academy/1995/inductionprobs.pdf

• We will see the Fibonacci numbers many times. We saw them today in covering a \(2 \times n\) rectangle with dominoes. Here is a nice occurrence of a related recurrence relation in gambling: the Double Plus One problem: https://www.youtube.com/watch?v=Esa2TYwDmwA

• We talked about tic-tac-toe today as a counting problem: how many `distinct' games are there. We are willing to consider games that are the same under rotation or reflection as the same game; see http://www.btinternet.com/~se16/hgb/tictactoe.htm for a nice analysis, or see the image here for optimal strategy. Chirality: http://www.rowland.harvard.edu/rjf/fischer/background.php

  • more on tic-tac-toe:
    • How many distinct games of tic-tac-toe are there? Do both in the case when we consider mirror images / flips to be the same and when we don't. Remember as soon as there are three in a row the game is over!
    • Redo the problem above, but now do it on a \(3 \times 3 \times 3\) tic-tac-toe board.
    • Redo the above problem but on a \(3 \times 3 \times \cdots \times 3\) board, where we have a total of \(n\) dimensions!
    • Interesting variants:
      • Ultimate tic-tac-toe: http://mathwithbaddrawings.com/2013/06/16/ultimate-tic-tac-toe/
      • Quantum tic-tac-toe: http://en.wikipedia.org/wiki/Quantum_tic-tac-toe
      • Bidding tic-tac-toe:
        • http://blogs.ams.org/mathgradblog/2011/02/25/gambling-tic-tac-toe/
        • http://apps.carleton.edu/curricular/math/assets/Comps11FinalPaperTTT.pdf
        • Discrete bidding games: http://arxiv.org/pdf/0801.0579v2
      • Gobblet tic-tac-toe: http://www.amazon.com/Blue-Orange-103-Gobblet-Gobblers/dp/B001TMXDMK  and  https://www.youtube.com/watch?v=Xpz7-8MbRvg
      • Probably the most famous movie occurrence of tic-tac-toe is from Wargames; the clip is here (the entire movie is online here, start around 1:44:17; this was a classic movie from my childhood).
  • Clip (joint with students from OIT) on Duality: https://www.youtube.com/watch?v=aMorr1h4Egs