Counting LEGO configurations

TL;DR

This paper develops algorithms to count and analyze the exponential growth of LEGO structure configurations, providing asymptotic estimates and explicit formulas for various dimensions and tile sizes.

Contribution

It introduces an algorithm for counting LEGO configurations and estimates their asymptotic growth, including explicit formulas for fixed tile counts.

Findings

Number of configurations grows exponentially with the number of tiles.

For 2D structures, growth is like A(w)μ(w)^n/n.

For 3D structures, growth is like Aμ^n/n^{3/2}, with μ ≈ 117.25.

Abstract

We discuss the problem of counting certain LEGO structures, primarily those comprising parallel $w \times 1$ tiles. These can be combined, as a single LEGO structure, by interlocking the tiles. %Alternatively, if the interlocking condition is relaxed, so that tiles can also be placed end-to-end, a greater number of possible configurations results. We also study the historically earlier problem of counting the number of ways to combine $2 \times 4$ LEGO tiles, which in this case gives a 3-dimensional structure. In all cases the number of configurations is dominated by an exponential growth term, $\mu^n$ where $n$ is the number of tiles. We present an algorithm for counting these various LEGO configurations, and use the data to estimate the asymptotics. We analyse the data so generated, and conjecture that, for the two-dimensional structures, the number of possible configurations grows like $A(w)\mu(w)^n/n,$ and we give numerical estimates for $A(w)$ and $\mu(w)$ for $w < 11,$ while for the three-dimensional structure the number of possible configurations is conjectured to grow like $A\mu^n/n^{3/2},$ where $\mu = 117.25 \pm 0.05.$ We also study the sequences that arise when we fix the number of tiles $n,$ and vary the tile size $w.$ We prove that the sequences are polynomials of degree $n-1,$ and we give these explicitly for $n=1 \ldots 14.$