The Mathematics of LEGO Bricks

The big build will be Monday Jan 15 (in Paresky), the Adventures in Learning are Fridays from 1:30 to 3:30ish at the Elementary School (starting 1/5)

Professor Steven Miller (Math/Stats); TAs Cameron and Kayla Miller (LEGO Grand Master Builders)

(translation generously done by Antony Kim: https://docs.google.com/document/d/1gVixldnb9FPOIumq6y2qBQhb9W--R8OuKUqMHXfD6Vk/edit)

LEGO COURSE: 2018: Opening Lecture: pptx  pdf (video here: https://youtu.be/qtXPyhmwwLw)

MLK BRIDGE BUILD: 2018: http://web.williams.edu/Mathematics/sjmiller/public_html/legos/mlkbridge/bridge2018/index.htm

Main goals:

• Fridays: Adventures in Learning: Outreach at Williamstown Elementary: Fridays 3-5pm

LEGO COURSE: 2017: MLK Bridge build: http://web.williams.edu/Mathematics/sjmiller/public_html/legos/mlkbridge/bridge2017/index.htm

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Links for 2016: Slides for First Day:  pptx    pdf

Link  to pictures of successful MLK Bridge

Videos from 2015: http://www.iberkshires.com/story/48408/Williams-Team-Breaks-Record-on-Lego-Star-Destroyer-Build.html (Jan 29, 2015)

Links on the successful build from 2014: http://math.williams.edu/lego-superstar-record/

2014: World Record Set! Under 10 1/2 minutes! Best videos are the first videos from each camera. While we didn't break our 10 minute goal, we broke several other targets (30 minutes, 20 minutes, 15 minutes, 12:34 minutes, 11:23 minutes or the Fibonacci time, and a time that rounds to 10 and not 11, also known as 10 1/2 minutes). We learned a lot, had a lot of fun, and are looking forward to our next project! Please email sjm1@williams.edu if you were here and participated (and not enrolled in the class) so we can properly credit you, or if you want updates about the LEGO club we're forming. Thanks to all for coming out and supporting us, for pitching in and helping. Special thanks to Dining Services and ECom for the festivities and funding help. It was great to see so many people from so many parts of the Williamstown Community come together, from faculty and staff to students and children. LEG GODT. --Steve (PS: We averaged a tad over 5.075 pieces per second, or about 304.54 pieces a minute! If we had averaged exactly 5 pieces a second the time would have been.... 10 minutes, 30.402 seconds! Looking at it that way, 10:30 is a pretty natural barrier, and we beat that! I feel a bit better.)

NEW: Practice videos:    Bag 1 (incomplete)        Bag 2        Bag 3        Bag 4        Bag 5

Click here for an expanded lectures page from 2014, which has links to the videos, articles and optional problems.

Since their introduction in 1949, LEGO bricks have challenged and entertained millions. In this course we’ll explore some of the connections between LEGO bricks, mathematics and popular culture. Topics include the following:

1. Given a collection of LEGO bricks, how many different structures may be built using only the standard snapping? The analysis requires us to develop some of the theory of combinatorics, and deal with the issue of two configurations that look different but are the same after standard moves (such as rotation, flipping about a line, and so on). We will use this problem as a springboard to study related issues in mathematics, especially in game theory.
2. Given a collection of LEGO bricks, how can you build desired objects? This ranges from building miniature replicas to functional items (which can now be done through special’ pieces).
3. The business model of the LEGO Group has changed greatly since the ‘40s and ‘50s. While they still hold their products to the highest standard, the generic themes (such as city and space) are now greatly supplemented by various alliances (Superheros, Star Wars, Harry Potter, Lord of The Rings, …). We will examine some business cases involving LEGO in order to get a sense of how companies determine priorities, including a discussion of the recent LEGO Friends line and gender issues.
4. One of the greatest computational advances is the ability to parallelize certain computations. Some programs must be run in order, where Step N cannot be done until Step N-1 is completed. Other problems, however, are such that multiple steps can be done simultaneously; examples include GIMPS (the Great Internet Mersenne Prime Search), SETI, mapping the human genome, factorizing numbers, and checking the Riemann Hypothesis. We will discuss the general theory of such computations and its effect on attacking important problems. We will implement our skills by parallelizing the building of the LEGO Star Wars Superstar Destroyer; as it is 3152 pieces, we see the need of having a good, efficient strategy if we are to complete it during the course! (Recent addition: the hope is to set a Guinness record -- I'll be in touch with them, and hopefully we'll have a fun event in Paresky; it's been suggested we buy TWO sets and have one to practice!)

http://cache.LEGO.com/bigdownloads/buildinginstructions/6005794.pdf (instructions for the set -- I think it's 75 megs!) (also here)

Evaluation will be based on class participation, the completion of problem sets involving the mathematical concepts, a final report on one of the topics, and adherence to Leg Godt’.

No prerequisites. Preference will be determined if needed by an application essay, interview and/or meeting.  Enrollment limit: 25 (ok, we're at 58!). Expected cost: \$50 for supplies.

Meeting Time: TBD. I will be away Jan 9 - 13, Jan 16 and Jan 26 onward. Meetings times will be split between mornings and afternoons (there will be some afternoon meetings so that the TAs, Cameron and Kayla, can participate).

Gamplan: In addition to general play (building sets, watching movies, playing LEGO Creationary).... Units where people need to volunteer are Unit 3 (presentation material related to the LEGO company), Unit 5 (volunteering in the elementary school), and Unit 4 (building the SuperStar Destroyer -- everyone will do this, but I need a lot of help parallelizing the construction).

• Unit 1: General theory of counting and combinatorics. This will lead to the first project: write a computer program to determine how many different structures one can create with a given set of bricks. Some care is required in defining what is allowable. This is an outstanding problem for people with computer expertise, and leads to discussions on recurrence relations and recursively defined functions. Thus in this unit you'll learn about binomial coefficients, Fibonacci numbers, the factorial function, and so on.

• Unit 2: Efficiency: How does one do something quickly? We'll talk about efficient algorithms throughout history. These include Babylonian multiplication (which leads to look-up tables), telescoping sums, the spectral theory of self-adjoint operators from linear algebra (providing a fast way of finding Fibonacci numbers), fast multiplication (which is a key ingredient in cryptography), fast matrix multiplication (you wouldn't believe how important this is), Horner's algorithm (used to allow us to make fractal pictures quickly), and culminating in parallel computing (with important networked applications such as the human genome project, SETI, the search for zeros of the Riemann zeta function, GIMPS, ...).

• Unit 3: General play, history of the company. Why is LEGO so successful? How has their strategy evolved over time? What different product lines have they used, what worked, what didn't? I would like several people to research these questions and report to the class. There is a lot that can be learned from LEGO's successes and failures, and some of the controversies (such as the reception the LEGO Friends line has received). Here's a recent article on them from Forbes: http://www.forbes.com/sites/quora/2013/08/02/why-was-2012-LEGOs-most-successful-year-in-terms-of-revenue/ . For those of you considering a career in business, this is a great topic to explore.

• Unit 4: The Construction: The culmination of the course is our attempt to set a world record in construction of a LEGO SuperStar Destroyer. Here's a clip of a team of 5 doing it in about 90 minutes (http://www.youtube.com/watch?v=HDIWqtnUGdo; here's a nice time-lapse: http://www.youtube.com/watch?v=R6Y08XoZgPo which really helps us see how things fit); I want to know how fast we can do it. Ideally I'd like to build it in 10 minutes or less. There are SEVEN pages, and over 200 pages of instruction (http://cache.LEGO.com/bigdownloads/buildinginstructions/6005794.pdf). I'll print out some copies of the directions -- we're going to really need to plan this! You'll be doing a lot of the disucssions among yourselves while I'm in Baltimore (the week of the 12th to 19th). I want to have at least 10 people per bag, which means 70 people. Ideally we'll have 15 per bag. We have about 30 in the class, and another 10 or so who are willing to help. If you have friends who are free and able to help a bit, please have them contact me. We'll have a big party / event when we do this in Paresky. That'll be either Thursday Jan 23rd or Friday Jan 24th. I need seven `Bag Captains'. Duties include being in charge of the parallelization of the bag and the assembly. I will be Supreme Commander, in charge of everything. If we are short on kids, the Children's Center is closed on the 24th, and we could get some 'young' assistants....

• Unit 5: Education outreach: An important part of life is giving back. This unit involves going to the Children's Center and interacting with the kids in the PreSchool II and SchoolAge classrooms with LEGOs. This can include the creationary game for SchoolAge, and helping them build and play a version of Connect4.

Videos of 2014 Lectures: An expanded lecture page, with more reading and problems to think about, is available here.

Here are some small ships I made. The Enterprise and the Bird of Prey came from leftover pieces from the Megablocks (gasp!) Enterprise D (after building these there was only one piece remaining); extra credit to whomever finds the asymmetry in the Bird of Prey. The X-Wing was done from the bucket of LEGO bricks I saved from my childhood.

The next picture is the LEGO/Twisty cake topping I made for my wedding -- I'll try to upload a better pic, the LEGO base was built sideways. The one after that is from Halloween '13; can you find the LEGO pumpkin?

It was supposed to be the German battleship Bismark (originally its main gun turrents had two guns instead of three), but I didn't have enough grey pieces to make the scale right, so in the end my kids and I turned it into a modified heavy cruiser. I'd like to get a huge number of general LEGO bricks and have us make a great creation of something for the class.

Quick links:    Gameplan        LEGO Links       Videos of Lectures    Some of my LEGO creations