Related papers: Counting LEGO configurations
We propose the further study of the rate of growth of the number of contiguous buildings which may be made from n LEGO blocks of the same size and color. Specializing to blocks of dimension 2x4 we give upper and lower bounds, and speculate…
We introduce a method to automatically compute LEGO Technic models from user input sketches, optionally with motion annotations. The generated models resemble the input sketches with coherently-connected bricks and simple layouts, while…
We tackle the problem of sequential brick assembly with LEGO bricks to create combinatorial 3D structures. This problem is challenging since this brick assembly task encompasses the characteristics of combinatorial optimization problems. In…
Structural stability is a necessary condition for successful construction of an assembly. However, designing a stable assembly requires a non-trivial effort since a slight variation in the design could significantly affect the structural…
In areas as diverse as contemporary art, play structures, climbing equipment, and modular construction toys, we see the presence of building block-like polyhedral complexes, which are generalizations of the pieces in the game Tetris. We…
Generative models are now used to create a variety of high-quality digital artifacts. Yet their use in designing physical objects has received far less attention. In this paper, we advocate for the construction toy, LEGO, as a platform for…
Given a tiling of a 2D grid with several types of tiles, we can count for every row and column how many tiles of each type it intersects. These numbers are called the_projections_. We are interested in the problem of reconstructing a tiling…
We train a language model to generate LEGO-brick build sequences. While prior work has been restricted to discrete, voxel-like towers, we consider a much broader set of pieces, encompassing thousands of part types with diverse connection…
Can you decide if there is a coincidence in the numbers counting two different combinatorial objects? For example, can you decide if two regions in $\mathbb{R}^3$ have the same number of domino tilings? There are two versions of the…
Visual understanding of geometric structures with complex spatial relationships is a fundamental component of human intelligence. As children, we learn how to reason about structure not only from observation, but also by interacting with…
We consider the well-studied pattern counting problem: given a permutation $\pi \in \mathbb{S}_n$ and an integer $k > 1$, count the number of order-isomorphic occurrences of every pattern $\tau \in \mathbb{S}_k$ in $\pi$. Our first result…
H. N. V. Temperley's method for counting vertically convex polyominoes is modified, generalized, and most importantly, programmed (in Maple).
Assembly planning is a difficult problem for companies. Many disciplines such as design, planning, scheduling, and manufacturing execution need to be carefully engineered and coordinated to create successful product assembly plans. Recent…
Many complex systems are modular. Such systems can be represented as "component systems", i.e., sets of elementary components, such as LEGO bricks in LEGO sets. The bricks found in a LEGO set reflect a target architecture, which can be…
We present a new teaching and outreach activity based around the construction of a three-dimensional chart of isotopes using LEGO$^{\circledR}$ bricks. The activity, \emph{Binding Blocks}, demonstrates nuclear and astrophysical processes…
Multi-step spatial reasoning entails understanding and reasoning about spatial relationships across multiple sequential steps, which is crucial for tackling complex real-world applications, such as robotic manipulation, autonomous…
Large language models (LLMs) are essential in natural language processing (NLP) but are costly in data collection, pre-training, fine-tuning, and inference. Task-specific small language models (SLMs) offer a cheaper alternative but lack…
Tile-based self-assembly systems are capable of universal computation and algorithmically-directed growth. Systems capable of such behavior typically make use of "glue cooperation" in which the glues on at least $2$ sides of a tile must…
We prove that the number of tile types required to build squares of size n x n, in Winfree's abstract Tile Assembly Model, when restricted to using only non-cooperative tile bindings, is at least 2n-1, which is also the best known upper…
We count tilings of the $n \times m$ rectangular grid, cylinder, and torus with arbitrary tile sets up to arbitrary symmetries of the square and rectangle, along with cyclic shifting of rows and columns. This provides a unifying framework…