Count all combinations of six 2x4 LEGO bricks
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README.md

All Combinations of Six 2x4 LEGO Bricks

Main project page | Findings | Full paper | Project on GitHub | Updates on Eurobricks

Motivation

Previous results count the ways that 2x4 LEGO bricks can be connected at straight angles.

Bricks connected at right angles

We seek to find the number of ways up to six bricks can be connected when using all possible angles.

Bricks connected at odd angles

Our Findings

For an explanation of the terms used in this section, please refer to the Terminology section below. This is a work in progress, so the results are still incomplete. The tables below shows the current findings. The numbers marked with an asterix (*) are not yet final.

Number of bricks 1 2 3 4 5 6
RC's (previous results) 1 24 1.560 119.580 10.166.403 915.103.765
Models (our main results) 0 0 0 1.144 213.221* 26.417.316*

See the detailed findings page for a breakdown of these numbers, or go directly to the models page so see the models. The full paper is currently under construction, and will contain all details of the theory behind the numbers, as well as the software used to find them.

Terminology

In this section we define the terms used in the table of our current findings above.

  • Brick: A brick is a standard "2 by 4" LEGO brick with 8 studs on top.
  • Configuration: A configuration is any number of bricks connected by their studs. See any picture on this page for examples of configurations.
  • Rectilinear configuration (RC): An RC is a configuration where all bricks are connected at right angles. Previous results count the exact number of RC's.
  • Model: Our main contribution is to count the number of models. A model is a set of configurations where any configuration in the set can be turned into any other configuration of the set simply by turning bricks where they are connected to other bricks using a single stud. Furthermore, there may be no RC in a model. The picture below shows various configurations belonging to the same model.

Models

We typically use colors to highlight the individual sub-configurations that can be turned. This is, however, only for illustrative purposes. Our results, as well as previous results, assume that all bricks have the same color and are thus interchangeable.

What now?

If you have read the terminology above and want to see more details, then the findings page is the place to continue. If you want to see the models that the software has found, then you might appreciate the models page. The full paper is currently under construction, and we are working on improving the algorithm to find models using the software. The GitHub project is open for ideas of improvement.