PreprintCC BY-NC-ND 4.0arXiv
Fourfold Is Enough: Connected LEGO Models from One Brick Type
Abstract
Can a LEGO model built from many kinds of rectangular bricks be rebuilt using only the familiar 2×4 brick? Enlarging every dimension by four makes the pieces fit, but they must also connect. We give a four-layer arrangement that joins every finite face-connected union of lattice cubes, using eight upright bricks per source cube, and show that no smaller universal enlargement works. A second question has an unexpected answer: if connectivity is set aside, the minimum integer scale is always $1$, $2$, or $4$, never $3$. The missing value follows from a checkerboard argument applied separately inside each enlarged cell.
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BibTeX
@misc{gans2026fourfold,
title = {{Fourfold Is Enough: Connected LEGO Models from One Brick Type}},
author = {Joshua S. Gans},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09205},
eprint = {2609.09205},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09205},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Fourfold Is Enough: Connected LEGO Models from One Brick Type
AU - Joshua S. Gans
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.09205
UR - https://arxiv.org/abs/2609.09205
AB - Can a LEGO model built from many kinds of rectangular bricks be rebuilt using only the familiar 2×4 brick? Enlarging every dimension by four makes the pieces fit, but they must also connect. We give a four-layer arrangement that joins every finite face-connected union of lattice cubes, using eight upright bricks per source cube, and show that no smaller universal enlargement works. A second question has an unexpected answer: if connectivity is set aside, the minimum integer scale is always $1$, $2$, or $4$, never $3$. The missing value follows from a checkerboard argument applied separately inside each enlarged cell.
ER -