There is an informative write-up at scientific american . It says that three researchers who worked under his supervision found irregularities in published data and then notified the head of department.

The authors cover the lawn with a triangular grid graph. Mowing at every Vertex is mowing the entire lawn. They say that finding "an efficient path is easily achieved by well-known computer search algorithms". With some simple search algorithm finding a reasonably good path may be simple but the problem of the optimal path can be very hard.
A perfect mowing mows at every vertex exactly once. The perfect mowing exists if there is a hamiltonian path in the triangular grid graph on the lawn. In general the hamiltonian path problem is NP-complete even on the triangular grid graph. However [1] states:

D is the linearly-convex hull of the Star of David. A polynomial time algorithm which is not exactly simple is available in [2]. It can be applied to solid grid graphs.
This approximately means if your lawn is not shaped like the Star of David and does not enclose any trees, bushes or ponds, you can implement the algorithm from [2] and get an perfect mowing path in polynomial time.
[1] Gordon, Orlovich, Werner. COMPLEXITY OF THE HAMILTONIAN CYCLE PROBLEM IN TRIANGULAR GRID GRAPHS
[2] W. Lenhart and C. Umans. Hamiltonian Cycles in Solid Grid Graphs