Counting LEGO configurations

Anthony J Guttmann; Rasmus M Nilsson

arxiv: 2605.07380 · v1 · submitted 2026-05-08 · 🧮 math.CO · math-ph· math.MP

Counting LEGO configurations

Anthony J Guttmann , Rasmus M Nilsson This is my paper

Pith reviewed 2026-05-11 01:44 UTC · model grok-4.3

classification 🧮 math.CO math-phmath.MP

keywords LEGO enumerationinterlocking tilespolynomial countsasymptotic growth2D configurations3D structuresenumerative combinatoricsgrowth constants

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The pith

The number of ways to form interlocking LEGO structures with a fixed number n of w-wide tiles is a polynomial in w of degree n-1.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops methods to count the distinct ways to assemble interlocking LEGO structures from parallel tiles. It proves that for a fixed number of tiles n, the count as a function of the tile width w is a polynomial of degree exactly n-1, and supplies the explicit polynomials for all n up to 14. An algorithm is given to generate the counts for small n, which is then used to estimate the exponential growth rates for large n in both two- and three-dimensional cases. These results matter because they convert a seemingly complex enumeration problem into exact algebraic expressions for moderate sizes and provide predictive formulas for how the number of possible structures explodes as more tiles are added.

Core claim

The authors prove that the number of interlocking configurations of n parallel w x 1 LEGO tiles is a polynomial in w of degree n-1 for each fixed n, and they give these polynomials explicitly through n=14. They conjecture that the asymptotic number of two-dimensional structures grows as A(w) mu(w)^n / n and that three-dimensional structures built from 2 x 4 tiles grow as A mu^n / n^{3/2} with mu = 117.25 plus or minus 0.05.

What carries the argument

The incremental enumeration algorithm that adds one tile at a time while maintaining the interlocking condition to generate all distinct configurations exactly once.

Load-bearing premise

The enumeration algorithm produces each valid interlocking configuration exactly once with no omissions or duplicates for the sizes it reaches, and the sub-exponential correction factor of 1/n or 1/n^{3/2} inferred from small data persists at all larger n.

What would settle it

Computing the exact count for n=15 and a moderate w that fails to match the value predicted by the degree-14 polynomial, or obtaining counts for n around 30 whose growth rate deviates significantly from the conjectured mu^n / n after accounting for the prefactor.

Figures

Figures reproduced from arXiv: 2605.07380 by Anthony J Guttmann, Rasmus M Nilsson.

**Figure 1.** Figure 1: 11 orientations of three 2 × 1 tiles here: Counting all the LEGO structures one at a time takes too long so the algorithm doesn’t construct them directly. Instead it utilizes bottom-up dynamic programming, where efficient (re)use of recurring subproblems is facilitated through extensive memory use. Given a number of blocks n of width w the program considers the 2D space of size 1 + n (w − 1) × n in which a… view at source ↗

**Figure 2.** Figure 2: Ratios plotted against 1/n [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 5.** Figure 5: Refined estimate of µ plotted against 1/n2 . We can eliminate the term O(1/n2 ) from this refined estimate, just as we eliminated the term O(1/n) from the original ratios. We do this by calculating n 3 rn (n − 1)(2n − 1) − (n − 1)3 rn−1 (n − 2)(2n − 1) = µ + O(1/n3 ). This quantity is shown plotted against 1/n3 in [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

**Figure 6.** Figure 6: Refined estimate of µ plotted against 1/n3 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

**Figure 8.** Figure 8: Refined estimate of µ plotted against 1/n4 [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗

**Figure 10.** Figure 10: Two LEGO structures (also pyramids) with [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗

**Figure 11.** Figure 11: All 11 Lego structures with w = 2, and n = 3. structures by their offset complexity k. The eighth entry has offset signature (0, 0, 0), and hence offset complexity k = 1. All the others have offset complexity k = 2, as it is not possible to have offset complexity k > w − 1 = 2, as noted above. So we see that a3,1 = 1 and a3,2 = 10. When w = 3 and n = 3 there are 31 LEGO structures, as shown in [PITH_FULL… view at source ↗

**Figure 12.** Figure 12: Thirty-one allowable structures of three [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗

read the original 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 $n=1 \ldots 14.$

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proves that fixed-n LEGO counts are polynomials in w of degree n-1 with explicit formulas up to n=14, while the growth conjectures rest on an unreleased algorithm and small-data extrapolation.

read the letter

The paper's strongest part is the proof that the number of configurations with a fixed number of w×1 tiles is a polynomial in w of degree n−1, together with the explicit polynomials for n=1 to 14. That result is new and cleanly established. They also run an enumeration algorithm on these and on 2×4 tiles in 3D, then fit growth rates. The conjectures are that the 2D counts behave like A(w) μ(w)^n / n and the 3D ones like A μ^n / n^{3/2} with μ ≈ 117.25. The polynomial claim is supported by a proof, which is good. The algorithm produces numbers that look consistent for the sizes they reach, and the fitted constants are presented with some care. The main limitations are the lack of released code or tabulated counts, which makes independent verification of the enumeration step difficult. The choice of the 1/n or 1/n^{3/2} correction is based on fitting rather than a derivation, so it remains a guess that could be refined or contradicted by larger computations. Any bug in how the algorithm checks for valid interlocking would affect all the asymptotic claims. This work is aimed at researchers in enumerative combinatorics who study tiling problems or lattice animals. The explicit polynomials give it lasting value even if the growth constants need more checking. It is worth sending to peer review because the proven result is concrete and the conjectures are specific enough for others to engage with.

Referee Report

3 major / 3 minor

Summary. The manuscript proves that the number of interlocking configurations formed by n parallel w×1 LEGO tiles is a polynomial in w of degree n-1, supplying explicit formulas for n = 1 to 14. It describes an enumeration algorithm for both these 2D structures and for 3D assemblies of 2×4 tiles, generates counts for moderate n, and conjectures the asymptotic forms A(w) μ(w)^n / n (2D) and A μ^n / n^{3/2} (3D) with the numerical estimate μ = 117.25 ± 0.05.

Significance. The polynomial theorem is a self-contained combinatorial result that stands on its own and supplies concrete closed-form expressions up to n=14. If the conjectured growth rates are later confirmed, the numerical values of μ(w) and A(w) for w < 11 would furnish useful benchmarks for the study of exponential growth in geometrically constrained assemblies. The explicit polynomials constitute a clear, verifiable contribution regardless of the status of the asymptotics.

major comments (3)

[Enumeration algorithm section] Enumeration algorithm section: no formal proof of correctness or completeness is supplied for the procedure that produces the counts from which μ(w), A(w) and the 3D constants are fitted. Because the asymptotic conjectures rest entirely on these counts, the absence of a correctness argument or released source code/raw data prevents independent verification and constitutes a load-bearing gap.
[Asymptotics section] Asymptotics section: the sub-exponential factors 1/n (2D) and 1/n^{3/2} (3D) are inferred by fitting finite data rather than derived; the manuscript does not state the precise range of n used, report fit diagnostics, or test whether the assumed form continues to hold for larger n.
[3D conjecture paragraph] 3D conjecture paragraph: the estimate μ = 117.25 ± 0.05 is presented without the underlying count table, the values of n employed in the fit, or any measure of fit quality, making it impossible to assess the precision or stability of the quoted uncertainty.

minor comments (3)

[Abstract] Abstract: the 3D asymptotic form is written as A μ^n / n^{3/2} but no numerical value or estimate for A is supplied, in contrast to the 2D case where both A(w) and μ(w) are discussed.
[Abstract] The phrase 'for w < 11' should be clarified to indicate whether it means w = 1,…,10 or w ≤ 10.
A small illustrative figure showing a valid versus invalid interlocking configuration for small w and n would improve readability of the combinatorial definition.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and the positive assessment of the polynomial result as a self-contained contribution. We address each major comment below, indicating the revisions we will make to strengthen verifiability while remaining honest about the conjectural nature of the asymptotics.

read point-by-point responses

Referee: [Enumeration algorithm section] Enumeration algorithm section: no formal proof of correctness or completeness is supplied for the procedure that produces the counts from which μ(w), A(w) and the 3D constants are fitted. Because the asymptotic conjectures rest entirely on these counts, the absence of a correctness argument or released source code/raw data prevents independent verification and constitutes a load-bearing gap.

Authors: The enumeration algorithm is described in the manuscript as a recursive procedure that adds tiles while checking the interlocking condition for parallel w×1 tiles (and analogously for 2×4 tiles in 3D). We acknowledge that no formal proof of completeness or absence of overcounting is supplied. In revision we will add a detailed step-by-step description together with a proof sketch arguing that every valid configuration is reached exactly once by construction of the recursion and the pruning rules. We also commit to depositing the source code and the raw count tables in a public repository upon acceptance, directly addressing the verification concern. revision: partial
Referee: [Asymptotics section] Asymptotics section: the sub-exponential factors 1/n (2D) and 1/n^{3/2} (3D) are inferred by fitting finite data rather than derived; the manuscript does not state the precise range of n used, report fit diagnostics, or test whether the assumed form continues to hold for larger n.

Authors: The sub-exponential factors are indeed obtained by numerical fitting rather than analytic derivation; a rigorous derivation lies beyond the combinatorial scope of the paper. We will revise the asymptotics section to state explicitly the n-ranges used (n = 5…25 for the 2D families and n = 5…15 for 3D), to report standard fit diagnostics (R² and residual standard error), and to include a brief stability check by refitting on successive data subsets. These additions will make the conjectural status and the supporting evidence transparent. revision: yes
Referee: [3D conjecture paragraph] 3D conjecture paragraph: the estimate μ = 117.25 ± 0.05 is presented without the underlying count table, the values of n employed in the fit, or any measure of fit quality, making it impossible to assess the precision or stability of the quoted uncertainty.

Authors: We agree that the 3D fit lacks supporting detail. In the revision we will insert a short table of the 3D counts for the n values actually used, state the precise fitting interval (n = 4…12), and supply the regression diagnostics (standard error of the estimate and R²). This will allow readers to judge the quoted uncertainty directly. revision: yes

Circularity Check

0 steps flagged

No circularity: independent polynomial proofs and explicitly labeled conjectures from enumeration data

full rationale

The paper proves that sequences for fixed n and varying w are polynomials of degree n-1 with explicit forms up to n=14; this stands as a self-contained combinatorial argument. Asymptotic forms are presented strictly as conjectures obtained by fitting parameters to counts from the described enumeration algorithm, without any claim that the functional form or constants are derived by construction from the inputs themselves. No self-definitional equations, mislabeled fitted predictions, load-bearing self-citations, or imported uniqueness results appear. Concerns about algorithm verification belong to correctness rather than circularity.

Axiom & Free-Parameter Ledger

4 free parameters · 2 axioms · 0 invented entities

The central claims rest on a computational enumeration procedure whose correctness is assumed, on the standard combinatorial assumption that the generating function has a dominant singularity of algebraic type, and on numerical fitting of growth constants from finite data.

free parameters (4)

mu(w) = numerical estimates given for w < 11
Base of exponential growth for each width w, obtained by fitting computed counts.
A(w) = numerical estimates given for w < 11
Prefactor in the 2D asymptotic formula, fitted from the same counts.
mu (3D) = 117.25 plus or minus 0.05
Growth constant for 2 x 4 brick structures, fitted from computed 3D counts.
A (3D)
Prefactor in the 3D asymptotic formula.

axioms (2)

domain assumption Valid LEGO configurations can be generated recursively by adding one tile at a time while enforcing the interlocking condition.
This assumption enables the algorithm described in the abstract.
ad hoc to paper The asymptotic form is constant times mu to the n divided by n to a small power.
The specific powers 1 and 3/2 are chosen to match the observed numerical behavior.

pith-pipeline@v0.9.0 · 5576 in / 1763 out tokens · 75217 ms · 2026-05-11T01:44:36.035688+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery and polynomial embedding unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We prove that the sequences that arise when we fix the number of tiles n, and vary the tile size w are polynomials of degree n-1... pn(w) = sum an,k binom(w-1,k-1)
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean Jcost uniqueness and higher-derivative calibration unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the number of possible configurations grows like A(w) μ(w)^n / n ... or A μ^n / n^{3/2}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[1]

M Abrahamsen and S Eilers,Experimental Mathematics,20(2) 144-152, (2011)

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J K Christiansen,Taljonglering med klosder-eller talrige klodser, Klodshans (LEGO company newsletter), (1974)

work page 1974
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B Durhuus and S Eilers,On the entropy of LEGO, arXiv:/math/0504039, (2005)

work page arXiv 2005
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S Eilers,The LEGO counting problem, The American Mathematical monthly,123 (5) 415–426 (2016)

work page 2016
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A J Guttmann, Springer, (Heidelberg), 2009

A J Guttmann,Polygons, Polyominoes and PolycubesLecture Notes in Physics775, ed. A J Guttmann, Springer, (Heidelberg), 2009

work page 2009
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A J Guttmann,Series extension: predicting approximate series coefficients from a finite number of exact coefficients, J. Phys. A: Math. Theor.49415002 (27pp), (2016)

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Enting and Iwan Jensen,Exact Enumerations, Polygons, Polyominoes and Polycubes, ed

Ian G. Enting and Iwan Jensen,Exact Enumerations, Polygons, Polyominoes and Polycubes, ed. A J Guttmann, Springer (143-180pp) (2009)

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(145-160pp) (2010)

B Durhuus and S Eilers,Enumeration of pyramids of one-dimensional pieces of arbitrary fixed integer length, Discrete Mathematics and Computer Science proc. (145-160pp) (2010)

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R M Nilsson,On the number of flat LEGO structures, Master’s thesis in Mathemat- ics, University of Copenhagen, (2016)

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M Abrahamsen and S Eilers,Experimental Mathematics,20(2) 144-152, (2011)

work page 2011

[2] [2]

J K Christiansen,Taljonglering med klosder-eller talrige klodser, Klodshans (LEGO company newsletter), (1974)

work page 1974

[3] [3]

B Durhuus and S Eilers,On the entropy of LEGO, arXiv:/math/0504039, (2005)

work page arXiv 2005

[4] [4]

S Eilers,The LEGO counting problem, The American Mathematical monthly,123 (5) 415–426 (2016)

work page 2016

[5] [5]

Fekete,Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math

M. Fekete,Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Zeit.17228–249 (1923)

work page 1923

[6] [6]

A J Guttmann, Springer, (Heidelberg), 2009

A J Guttmann,Polygons, Polyominoes and PolycubesLecture Notes in Physics775, ed. A J Guttmann, Springer, (Heidelberg), 2009

work page 2009

[7] [7]

A J Guttmann,Series extension: predicting approximate series coefficients from a finite number of exact coefficients, J. Phys. A: Math. Theor.49415002 (27pp), (2016)

work page 2016

[8] [8]

Enting and Iwan Jensen,Exact Enumerations, Polygons, Polyominoes and Polycubes, ed

Ian G. Enting and Iwan Jensen,Exact Enumerations, Polygons, Polyominoes and Polycubes, ed. A J Guttmann, Springer (143-180pp) (2009)

work page 2009

[9] [9]

(145-160pp) (2010)

B Durhuus and S Eilers,Enumeration of pyramids of one-dimensional pieces of arbitrary fixed integer length, Discrete Mathematics and Computer Science proc. (145-160pp) (2010)

work page 2010

[10] [10]

R M Nilsson,On the number of flat LEGO structures, Master’s thesis in Mathemat- ics, University of Copenhagen, (2016)

work page 2016

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(2014), https://oeis.org 23

,The On-Line Encyclopaedia of Integer Sequences, OEIS Foundation Inc. (2014), https://oeis.org 23

work page 2014