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For $r\geq1$, the $r$-neighbour bootstrap process in a graph $G$ starts witha set of infected vertices and, in each time step, every vertex with at least$r$ infected neighbours becomes infected. The initial infection percolates ifevery vertex of $G$ is eventually infected. We exactly determine the minimumcardinality of a set that percolates for the $3$-neighbour bootstrap processwhen $G$ is a $3$-dimensional grid with minimum side-length at least $11$. Wealso characterize the integers $a$ and $b$ for which there is a set ofcardinality $\frac{ab+a+b}{3}$ that percolates for the $3$-neighbour bootstrapprocess in the $a\times b$ grid; this solves a problem raised by Benevides,Bermond, Lesfari and Nisse [HAL Research Report 03161419v4, 2021].

Extremal Bounds for Three-Neighbour Bootstrap Percolation in Dimensions Two and Three