I saw the following problem in Mathematical Puzzles from Peter Winkler (very good book, by the way): imagine you infect k cases of a chessboard nxn and the infection spreads to a case if it has at least two neighbors infected. Then k = n is the minimum number such that it is possible to infect the whole chessboard.
I would like to know if it is known what happen in higher dimensions and other values for the number of neighbors for a case to be infected?



Infected square
There is an d-dimensional version of this problem in *The Art of Mathematics - Coffee Time in Memphis* by Bela Bollobas. (Problem 35) 


According to it, the answer is $k = \lceil d(n-1)/2 \rceil + 1$. 


