The Mathematics of LEGO Bricks

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

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

LEGO COURSE: 2023:

Class 01: 1-3-23: Introduction: Slides and Video https://youtu.be/vXV0GJm-Fnk

LEGO COURSE: 2022:

Class 01:1-4-22: Mechanics of Speed Build, Mathematics of Efficiency, Symmetry: https://youtu.be/WvYWhPToJF8

Class 02: 1-5-22: Dividing Sets: https://youtu.be/MI2ZP06WzEA

Class 03: 1-6-22: Bag Times: https://youtu.be/-KZTC97RTbQ

Class 04: 1-24-22: Theory Lecture: https://youtu.be/pzpVtsDsxRQ

LEGO COURSE: 2020: Opening Lecture: https://youtu.be/iTApYbKg0lA (slides here)

LEGO COURSE: 2019: Opening Lecture: https://youtu.be/ZzgmK8GSyIo (see 2018 for a more extensive opening lecture)

Slides: powerpoint    pdf    (lecture on Lego and counting, marketing, biology, ...): video here: https://youtu.be/0aWr-cvWTGI (issue with audio: uploaded audio to https://youtu.be/qCuE3GMUEGo so just have both playing at the same time!)

This year we'll be building the Apollo rocket to commemorate the 50th anniversary of the first manned landing on the moon. Click here for the set, or click here for the instructions. Berkshire Eagle story here: https://www.berkshireeagle.com/stories/having-a-blastoff-college-elementary-school-teams-race-to-build-lego-saturn-v,562807

• Apollo Saturn V Rocket: (All): Bag1    Bag2    Bag3    Bag4    Bag5    Bag6    Bag7    Bag8    Bag9    Bag10    Bag11    Bag12

• 3-in-1 Shuttle Creator: (All): Shuttle    Front Car    Back Car

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

• Mon Jan 15: Lego Bridge across 2nd floor of Paresky: Pictures here

  • Bridge build: Part I: https://youtu.be/9qxnbuIpUz8  Part II: https://youtu.be/sseblwTH23w  Part III: https://youtu.be/2dUBPJSS5xY  Part IV: https://youtu.be/Rf4q7RIU3qQ

• Wed Jan 17: Visit of Michael Uslan (Executive Producer of Batman movies)

  • Graduation speech: http://www.graduationwisdom.com/speeches/0018-uslan.htm

   

• Friday, Jan 19: Counting Brick Configurations, Intro to Games: Lecture: https://youtu.be/1KBRtHZBz78

  • Reading: 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, and 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.

  • Introduction to Ramsey theory here, and a link to Wikipedia on Ramsey's Theorem here.
  • Combining Lego bricks: https://arxiv.org/pdf/math/0504039.pdf and especially for six 2x4: http://www.math.ku.dk/~eilers/lego.html

   

• Wednesday, Jan 24: Counting Bricks, Double Plus One
  • https://youtu.be/YmDciAOw2JQ

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

• Lecture 1: Jan 3, 2017: Slides for First Day:  pptx    pdf     video: https://youtu.be/FRIxxSOyDKA
  • Read the following for Monday Jan 9; if you missed the first day of class you must also solve the problem mentioned in the video for Mon Jan 9.

    • Reading: Chirality: http://www.rowland.harvard.edu/rjf/fischer/background.php

    • Reading: 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, and 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.

  • Google sheet to sign up for areas of interest: https://docs.google.com/a/williams.edu/spreadsheets/d/19LBfUhTeHHK9imoS5SRppCHZFAkEDmS65SFHyu5bpQE/edit?usp=sharing_eixpa_nl&ts=586c8fab

  • LEGO Digital Designer: http://ldd.lego.com/en-us/

  • LEGO Idea: Pitch that reach 10,000: Voltron: https://ideas.lego.com/projects/138271 (should look at other projects to see what is successful)

   

• Lecture 2: Jan 9, 2017: video: https://youtu.be/oTURS7r2X68
  • Topics covered: Babylonian Mathematics, Look-up Tables, Ramsey Theory, Duality. If you want to learn more about any of these, please let me know. I'll include a link to Wikipedia's introduction to Ramsey theory here, a link to Wikipedia on Ramsey's Theorem here, and my youtube video on duality and the chess problem (warning: it does have the solution).

• Lecture 3: Jan 18, 2017: video: https://www.youtube.com/upload
  • Catalan program: code (Mathematica notebook)        printed pdf file
  • Pascal triangle example:    PascalTriangleMod2_Streamlined_StevenJMiller.nb        pascaltrianglemod2to2000ver2.dat        pascaltrianglemod2to4000ver2.dat    video here
  • Marriage problem:  cardgamemarriage.nb
  • Practice coding lectures: From Math/Stat 341: Probability            From Math 12: Mathematics of Lego Bricks
  • Practice coding problems and solutions: BasicCoding.nb    BasicCoding.pdf
  • OEIS:
  • Catalan numbers

• Lecture 4: Video: https://youtu.be/u_zVdFG_CxA
  • https://www.wired.com/2012/01/the-mathematics-of-lego/
  • skim, do not read unless you want: http://www.changizi.com/org.pdf
  • read: http://www.math.ku.dk/~eilers/lego.html
  • skim: https://arxiv.org/pdf/math/0504039.pdf
  • lying with statistics: https://www.youtube.com/watch?v=G0ZZJXw4MTA: This builds on our discussion of what it means to say you've won 3 of the last 4.

   

• Lecture 5: Video: https://youtu.be/98n_uZi80KM
  • Quadratic formula:
  • Cubic Formula:
  • Quartic Formula:
  • Abel-Ruffini Theorem:
  • Pigeon-hole Principle:
    
     

Links for 2016: Slides for First Day:  pptx    pdf       

Link  to pictures of successful MLK Bridge

• Lego suspension bridge: https://www.flickr.com/photos/suspensionstayed/sets/72157610808577323/with/3090342269/

  • http://lego.gizmodo.com/5272536/46-foot-long-self-supporting-lego-bridge-to-set-new-world-record/
  • http://www.ericharshbarger.org/lego/golden_gate_bridge.html
  • Google docs on ideas: https://docs.google.com/document/d/1AAqR2vjuWtMk-eQgv_YA-YOTSdZpOJ2axFsjA355HZs/edit 

• Lego Cota Tower

  • subpage: http://web.williams.edu/Mathematics/sjmiller/public_html/legos/cotatower/
  • related product: Lego speed champions: http://www.lego.com/en-us/speedchampions

• Lego base plates: http://www.ebay.com/bhp/lego-road-plates

• Lego base plates: http://shop.lego.com/en-US/Straight-Crossroad-Plates-7280?fromListing=listing

• Lego base plates: http://shop.lego.com/en-US/T-Junction-Curved-Road-Plates-7281?fromListing=listing

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/

• Lecture 1.1 (January 5, 2015): Introduction and class mechanics:  https://www.youtube.com/watch?v=0jFLfIhlwdU (unfortunately audio only worked for first 10 minutes)

• Lecture 1.2 (January 5, 2015): From Lego Bricks to Math (tic-tac-toe, metrics, chirality, statistics, ...): http://youtu.be/GZOvuaQykME

• Lecture 2.1 (January 6, 2015): Math Lecture 2: Efficiencies (Horner's algorithm, combinatorics): http://youtu.be/f3AujzMchLc

• 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

• Lecture 4.1 (January 8, 2015): Midway, Qwerty, Pascal mod 2: 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).

Additional Comments from 2015: Click here

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.)

• AWESOME time lapse video from 2014: http://www.youtube.com/watch?v=IpSjAYVZFBs&feature=youtu.be (10 minutes, 21 seconds)

• Camera 1: Video of record attempt            Video  of second attempt (failed piece, didn't break our record).

• Camera 2: Video of record attempt            Video  of second attempt (failed piece, didn't break our record).

• Video from Channel 13 from Albany: http://wnyt.com/article/12415/?vid=4897911&v=1  (video from Channel 13 of Albany's story on us, recorded  from my camera, downloadable here)

• Stories from the Berkshire Eagle: Before the event        After the event  (fixed links:   before    after)

• A shorter video here: http://www.youtube.com/watch?v=G47_AmfSOhc&feature=youtu.be

• Pictures from the class available here.

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.

Interesting LEGO pages / Reading:

• LEGO's home: http://www.LEGO.com/en-us/default.aspx

• Building LEGO models: http://LEGO.gizmodo.com/mad-genius-develops-LEGO-algorithm-to-build-any-3d-obje-647181411

• Life-size X-wing: http://LEGO.gizmodo.com/this-incredible-full-scale-LEGO-x-wing-is-the-largest-m-509484787

• xkcd strip: http://xkcd.com/1281/

• Ohio Stadium: http://www.odditycentral.com/news/fan-builds-11000-scale-LEGO-replica-of-ohio-stadium.html

• Oyo Toys (like LEGOs) commemorate the Red Sox '13 World Series Victory: http://www.youtube.com/watch?v=zCjL5Vr5Q5s

• Tortoise and prosthetic LEGOs: http://www.huffingtonpost.com/2013/11/08/tortoise-LEGO-wheel_n_4239566.html

• LEGOland (Westchester, NY): http://www.LEGOlanddiscoverycenter.com/westchester/

• LEGO Movie Trailer: http://www.youtube.com/watch?v=fZ_JOBCLF-I (comes out around February 7, 2013 I believe)

• LEGO records: http://www.recordholders.org/en/list/LEGO.html

• LEGO car (works and can get to 30kph): http://www.theregister.co.uk/2013/12/19/LEGO_car_powered_by_air_hits_32_kmh/

• LEGO art: http://brickartist.com/category/gallery/

• Articles

  • The mathematics of LEGO: 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, and LEGO: M. A. Changizinw, M. A. McDannaldwand D. Widdersw, J. Theor. Biology (2002) 218, 215–237).

  • Science article where LEGOs are mentioned.

  • Chirality: 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).

   

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

Below are some of my recent LEGO creations.

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