The 3-letter answer: ODD
Trace ODD from row 2, column 3. Move up to row 1, column 3, then turn left and finish at row 1, column 2. The letters read O → D → D. For a numbered check: 1 at row 2, column 3; 2 at row 1, column 3; 3 at row 1, column 2.
This path uses one upward move and one leftward move, with one turn and two straight segments. It spans two rows and two columns, visits the board edge twice, and does not enter a corner. Its starting cell has two open orthogonal neighbors, while its ending cell has three. Completing it assigns 3 cells and leaves 18 open cells for the longer answers.