Additional Comments for Lego Winter Study: January 2015 (see also http://math.williams.edu/lego-superstar-record/)

• Lecture 1: January 5, 2015: We started the day with some course mechanics, talked about the goals, and ended by using Lego bricks as a springboard to math.
  • Slides for first day: powerpoint        pdf
  • Video for lecture 1.1 (course mechanics and goal):  https://www.youtube.com/watch?v=0jFLfIhlwdU  (unfortunately audio only worked for first 10 minutes)
  • Video for lecture 1.2 (tic-tac-toe, metrics, Laffer curve, ...): http://youtu.be/GZOvuaQykME
  • Topics to think about:
    • 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
    • Write a program to figure out how many ways to combine \(n\) bricks, where all bricks are the same. Do this for bricks that are \(m \times n\) for various \(m\) and \(n\).
      • We talked about chirality and mirror images in biology; there are a lot of great articles on this. Look at http://www.rowland.harvard.edu/rjf/fischer/background.php .
      • Related to this is a wonderful story by Isaac Asimov, Mirror Image (click on the pdf file and search for Mirror Image to get to the story).
    • Read about solutions to a Rubik's cube to see more about symmetries.
    • Statistics.
      • We also discussed scaling issues with LEGO sets, specifically how the number of pieces grows as a function of the set. There are lots of great reads, relating this to biological and other complexities. 
        • Go to Wired: http://www.wired.com/wiredscience/2012/01/the-mathematics-of-LEGO/ (BY SAMUEL ARBESMAN 01.06.12).
        • Scaling and LEGO: http://www.changizi.com/org.pdf (Scaling of Differentiation in Networks: Nervous Systems, Organisms, Ant Colonies, Ecosystems, Businesses, Universities, Cities, Electronic Circuits.
        •  LEGO: M. A. Changizinw, M. A. McDannaldwand D. Widdersw, J. Theor. Biology (2002) 218, 215–237). See also Science article where LEGO bricks are mentioned.
      • Related to the above: look at the price of different LEGO sets as a function of the number of pieces and what line it is (general city, Star Wars, Harry Potter, Lone Ranger, LEGO Friends, ...). Try to find relationships (if you know regression here's a terrific place to use it!). Often taking a logarithm and doing a log-log plot is useful as it converts power relationships to linear ones.
      • A large part of the first day was concerned with determining the right statistic. For example, if we say a team has won 3 of the last 4 it sounds like they're hot (with a winning percentage of .750); however, the number is misleading. In their last 5 games they must have lost game 5, else we would have said they've won 4 of the last 5. Similarly we know game 4 must have been a win, or we would say they've won 3 in a row. Thus they are 3-2 in the last 5, and a little work shows they are 2-1 in the last 3. These are close to what you'd expect for an average team.
      • h-index: http://en.wikipedia.org/wiki/H-index
    • We ended with a brief analysis of the Laffer curve. A lot of our analysis was similar to what happened when we looked at whether or not it's better to have lots of little pieces or a small number of large pieces (in terms of getting the most combinations). Frequently extreme cases can help and give us a clue. For the Laffer curve, we analyzed what happens with a tax rate of 0% or 100%, and used that to get a sense of the intermediate behavior. For looking at combinations of bricks, if each were 1x1 there would only be one way to do; conversely if there is just one large brick there is only one possibility; we thus see that, similar to the Laffer curve, the optimum must be somewhere between the extremes.
    • Lego links:
      • Sean Kenney pages: http://www.seankenney.com/portfolio/times_square/
      • Buying individual pieces: http://www.seankenney.com/shop/bricks/
  • Lecture 2: January 6, 2015: We discussed efficiency of computation, seeing that many processes that we are used to can be done significantly faster than we are used to.
    • Video online here: Math Lecture 2: Efficiencies (Horner's algorithm, combinatorics): http://youtu.be/f3AujzMchLc
    • 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; 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.
    • There are many processes in mathematics that run in far less time than they appear. Below is a quick description of a few, many of which you may ha've already seen (but this is a good time to revisit). The Euclidean algorithm is a terrific example. It runs far faster than expected; it doesn't take \(\min(x,y)\) steps but rather \(2 \log(\min(x,y))\). It's a beautiful example of a major theme of the course, the need to do things fast.
      • Horner's algorithm to evaluate a polynomial quickly: \(4x^3 + 5x^2 - 3x + 8 = ((4x+5)x - 3)x + 8\) (saves a few multiplications!). Saving multiplications is very important; one application of evaluating polynomials quickly is in constructing Mandelbrot sets (see below when we discuss Newton's method).
        • The best known algorithm is the Coopersmith-Winograd algorithm, which is of the order of \(N^{2.376}\) multiplications. See also this paper for some comparison analysis, or email me if you want to see some of these papers.
          • Some important facts. The Strassen algorithm has some issues with numerical stability.
          • One can ask similar questions about one dimension matrices, ie, how many bit operations does it take to multiply two N digit numbers. It can be done in less than N^2 bit operations (again, very surprising!). One way to do this is with the Karatsuba algorithm (see also the Mathworld entry for the Karatsuba algorithm).
      • Telescoping sums: often re-arranging the algebra leads to a significantly easier computation.
      • Fast matrix multiplication: Naively we expect it takes \(N^3\) multiplications to find all \(N^2\) entries of \(A^2\) or \(AB\) when \(A\) and \(B\) are \(N\times N\) matrices. The Strassen algorithm (see also the Mathworld entry here, which I think is a bit more readable) does it in about \(N^{\log_2 7}\); the reason for this savings is that they can multiply two \(2\times 2\) matrices with \(7\) and not \(8\) multiplications (\(3 = \log_2 8\)). The best known algorithm is the Coopersmith-Winograd algorithm, which is of the order of \(N^{2.376}\) multiplications. See also this paper for some comparison analysis, or email me if you want to see some of these papers.
        • Some important facts. The Strassen algorithm has some issues with numerical stability.
        • One can ask similar questions about one dimension matrices, ie, how many bit operations does it take to multiply two \(N\) digit numbers. It can be done in less than \(N^2\) bit operations (again, very surprising!). One way to do this is with the Karatsuba algorithm (see also the Mathworld entry for the Karatsuba algorithm).
      •  Newton's method is significantly more powerful than divide and conquer (also called the bisecting algorithm); this is not surprising as it assumes more information about the function of interest (namely, differentiability). The numerical stability of Newton's method leads to many fascinating problems. One terrific example is looking at roots in the complex plane of a polynomial. We assign each root a different color (other than purple), and then given any point in the complex plane, we apply Newton's method to that point repeatedly until one of two things happen: it converges to a root or it diverges. If the iterates of our point converges to a root, we color our point the same color as that root, else we color it purple. This leads to Newton fractals, where two points extremely close to each other can be colored differently, with remarkable behavior as you zoom in. If you're interested in more information, let me know; a good chaos program is xaos (I have other links to such programs for those interested). One final aside: it is often important to evaluate these polynomials rapidly; naive substitution is often too slow, and Horner's algorithm is frequently used. What we're really doing in Newton's method is replacing a complicated function with a linearization.
        • One of the most important applications of the Euclidean algorithm is in RSA (cryptography) to compute modular inverses, and to determine some of the items needed for our keys.
        • There were several great questions about the efficiency of the Euclidean algorithm. The straightforward bound shows it must run in at most \(x\) steps (where \(x < y\) are our two numbers). After some analysis it can be showed it runs in at most \(2 \log_2(x)\) steps; however, maybe it's even faster, or faster on average. Click here for more on it's runtimes, and the various measures (average run time, worst case scenario). This leads to the topic of computational complexity.
    • We then turned to 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 talked a bit about counting 'good' games in tic-tac-toe, and ignoring counting obviously bad ones. (Consider the error in the classic Princess Bride `Chess' Match or in the Princess Bridge Battle of Wits). 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 (how many words can we form from Yale versus Mississippi). 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 x1 + ... + x5 = 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 x1^k + ... + xs^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).
  • Lecture 3: January 7, 2015: We talked  about ways to quickly check for errors, with applications  to the build.

    • Lecture 3.1 (January 7, 2015): Van Halen and Brown M&Ms, Telescoping Sums, Check digits: http://youtu.be/_Q_AKCU0xPk

    • Lecture 3.2 (January 7, 2015): Opening up the box: http://youtu.be/ftCWQzZ295E

    • Van Halen and Brown M&Ms:
      • Jump: https://www.youtube.com/watch?v=iKSyYGJHOGA
      • No browns as a quick check to make sure contracts are read: http://www.snopes.com/music/artists/vanhalen.asp
      • From snopes.com: The legendary "no brown M&Ms" contract clause was indeed real, but the purported motivation for it was not. The M&Ms provision was included in Van Halen's contracts not as an act of caprice, but because it served a practical purpose: to provide a simple way of determining whether the technical specifications of the contract had been thoroughly read and complied with.  Read more at http://www.snopes.com/music/artists/vanhalen.asp#7vzAFS4rS0UlMGQ5.99
    • Telescoping sums: often re-arranging the algebra leads to a significantly easier computation.
    •  Check digit schemes.

      • Bar codes

      • Check digit schemes

      • Universal Product Code (25th anniversary article)

    • Modular arithmetic:

    • Babylonian mathematics: There's an even better way to multiply then I covered; click here. The idea here is the precursor to the look-up tables we use daily. Click here for more on base 60.

    • \(\pi\) versus \(\tau\): the \(\tau\) manifesto and the \(\pi\) manifesto.

  • Lecture 4: January 8, 2015: We concluded the math lectures today.
    • Video online here: http://youtu.be/lRYN5y_BI6M (unfortunately the Mathematica video doesn't display  well, so you need to see https://www.youtube.com/watch?v=tt4_4YajqRM).
    • Qwerty keyboards, and Dvorac keyboards:

    • The Battle of Midway: My favorite part is how we figured out the target was Midway (click this link).

    • The binomial coefficients are the entries in Pascal's triangle. If you look at these coefficients modulo 2, something interesting appears, and you're on your way to fractal geometry! Video from my Mathematica notebook is here: https://www.youtube.com/watch?v=tt4_4YajqRM. There is a huge theory about Pascal's triangle modulo different integers. There are numerous resources; a quick search gave the following among others:
      • http://orion.math.iastate.edu/reu/modupasc.htm
      • Java program to visualize: http://math.fau.edu/richman/mla/pascal.htm
      • One of many papers....
      • For more on dimensionality, Hausdorf dimensions: http://en.wikipedia.org/wiki/Hausdorff_dimension
      • Pascal's Triangle and Binomial Coefficients (http://youtu.be/nrknXC8xmTU): 41minutes: goes through Pascal's triangle, what it is and why it works, why the relations hold, and shows how to get from that to Chaos Theory and Fractals (we can discuss more on this later). Includes a Mathematica video generating the triangle modulo 2, discussions on related problems, and issues in computing these triangles quickly. Gives story proofs of some binomial identities. (This is from a lecture I gave to middle and high school teachers in 2014.)