Answer 1: the four-letter path
Trace AREA from row 6, column 5. Move up to row 5, column 5, turn right to row 5, column 6, and finish by moving up to row 4, column 6. The cells read A → R → E → A. This compact path uses two upward steps and one rightward step, changing direction twice across three straight segments.
For a position-by-position check: 1 at row 6, column 5; 2 at row 5, column 5; 3 at row 5, column 6; 4 at row 4, column 6. Its starting and ending cells each have two open orthogonal neighbors. The route visits three edge cells, no corner cells, and spans three rows by two columns. Placed first, it assigns four cells and leaves eighteen open cells unassigned for the longer entries.