Problem #PRU-100482

Problem

A young mathematician had quite an odd dream last night. In his dream he was a knight on a $4\times 4$ board. Moreover, he was moving like a knight moves on the usual chessboard. In the morning he could not remember what was actually happening in his dream, though the young mathematician is pretty sure that either

(a) he has passed exactly once through all the cells of the board except for the one at the bottom leftmost corner, or

(b) he has passed exactly once through all the cells of the board.

For each possibility examine if it could happen or not.