Twin jumping checkers in Zd

We consider a generalization of an old checker jumping problem from d = 2 to d ⩾ 2: What is the maximum of xd if it is possible to bring a checker to the point (x1, x2, … xd) in Zd, starting with a distribution of checkers at lattice-points in the half-space xd ⩽ 0? We prove that the answer is 3d − 2. The next question is as follows: Bringing two checkers to the level 3d − 2, what is the minimum distance between them? We prove that the answer is 3. This was not known before, even when d = 2.