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:



Class 01: 1-3-23: Introduction: Slides and Video



Class 01:1-4-22: Mechanics of Speed Build, Mathematics of Efficiency, Symmetry:

Class 02: 1-5-22: Dividing Sets:

Class 03: 1-6-22: Bag Times:

Class 04: 1-24-22: Theory Lecture:


LEGO COURSE: 2020: Opening Lecture: (slides here)

LEGO COURSE: 2019: Opening Lecture: (see 2018 for a more extensive opening lecture)

Slides: powerpoint    pdf    (lecture on Lego and counting, marketing, biology, ...): video here: (issue with audio: uploaded audio to 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:,562807


LEGO COURSE: 2018: Opening Lecture: pptx  pdf (video here:


Main goals:


LEGO COURSE: 2017: MLK Bridge build:

Links for 2016: Slides for First Day:  pptx    pdf       

Link  to pictures of successful MLK Bridge


Videos from 2015: (Jan 29, 2015)

Links on the successful build from 2014:

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 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!) (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).

Interesting LEGO pages / Reading:

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