For r≥1, the r-neighbour bootstrap process in a graph G starts with
a set of infected vertices and, in each time step, every vertex with at least
r infected neighbours becomes infected. The initial infection percolates if
every vertex of G is eventually infected. We exactly determine the minimum
cardinality of a set that percolates for the 3-neighbour bootstrap process
when G is a 3-dimensional grid with minimum side-length at least 11. We
also characterize the integers a and b for which there is a set of
cardinality 3ab+a+b​ that percolates for the 3-neighbour bootstrap
process in the a×b grid; this solves a problem raised by Benevides,
Bermond, Lesfari and Nisse [HAL Research Report 03161419v4, 2021].Comment: 45 page