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 $2\times4$ 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.