Geometric probabilities for cubic lattices with cubic obstacles

In this paper we present solutions for problems of
Buffon type for a cubic lattice R(L, a) consisting of cubic obstacles
with edges 2a, having as symmetry center the points Mh,k,l =
(hL, kL, lL), h, k, l ∈ Z and the faces parallel to the coordinate
planes and the lattice R0(L, a) obtained by R(L, a) adding the
plane portions delimited by the following segments: {(x, kL, lL) :
x ∈ [hL+a, (h+1)L−a]}, {(hL, y, lL) : y ∈ [hL+a, (h+1)L−a]},
{(hL, kL, z) : z ∈ [hL + a, (h + 1)L − a]}, h, k, l ∈ Z.