Ideals I, J ⊂ k[x_0,...,x_n] are directly Gorenstein linked if there is a Gorenstein ideal K ⊂ I ∩ J such that K:I = J and K:J = I. The equivalence relation - Gorenstein linkage - generated by such direct linkages turns out to be very useful for the studying curves in P^3, but its significance is still not at all clear in codimension > 2. In 2001 Hartshorne proposed the problem of determining whether the ideal of a set of 20 general points in P^3 is Gorenstein-linked to a complete intersection. In November, Hartshorne, Schreyer and I were able to determine the graph of all direct Gorenstein linkages between general sets of points in P^3. Computer algebra, used in a somewhat novel way, plays an essential role in the proof. I will describe the background of the theory and explain some of the ideas of the proof.