An ant is at the bottom-left corner of a grid (point P) as shown above. It aims to move to the top right corner of the grid. The ant moves only along the lines marked in the grid such that the current distance to the top-right corner strictly decreases. Which one of the following is a part of a possible trajectory of the ant during the movement?

Correct Option: The correct option is the second one, which shows a staircase-like path moving only upwards and rightwards from point P.

To understand why this is the correct trajectory, let us define a coordinate system for the grid:
Let the bottom-left corner (point P) be the origin $(0,0)$, and let the top-right destination corner be $(4,4)$.
At any point $(x,y)$ on the grid, the straight-line (Euclidean) distance to the destination is given by the distance formula:

$$d=\sqrt{{(4-x)}^2+{(4-y)}^2}$$

According to the problem, the ant can only move along the grid lines such that its distance to the top-right corner strictly decreases at every step. Let us analyze how each direction of motion affects the distance:
1. Moving Right: Moving from $(x,y)$ to $(x+1,y)$ decreases the value of $(4-x)$, thereby reducing the distance $d$.
2. Moving Up: Moving from $(x,y)$ to $(x,y+1)$ decreases the value of $(4-y)$, thereby reducing the distance $d$.
3. Moving Left: Moving to $(x-1,y)$ increases the distance $d$.
4. Moving Down: Moving to $(x,y-1)$ increases the distance $d$.

Thus, any valid trajectory for the ant must consist strictly of rightward and upward steps. Any step to the left or downward will increase the distance to the destination, violating the condition.

Now let us evaluate the given options based on this rule:
- The first option shows a closed rectangular loop starting from P (UP, RIGHT, DOWN, LEFT), which includes DOWN and LEFT movements, making it invalid.
- The second option (correct answer) shows a path starting at P that goes UP, RIGHT, UP, and RIGHT. Since all segments of this path are directed only UP or RIGHT, the distance to the top-right corner strictly decreases at every step.
- The third and fourth options both contain downward segments, which increase the distance and are therefore invalid.