Problem Analysis

The ant is at point \( P \) (bottom-left corner) and wants to reach the top-right corner of a grid. Let's define the coordinates of \( P \) as \( (0, 0) \) and the top-right corner as \( (4, 4) \) based on the \( 4 \times 4 \) cell grid shown.

The condition for the ant's movement is that the current distance to the top-right corner strictly decreases. Let the current position of the ant be \( (x, y) \). The distance to the target \( (4, 4) \) can be considered as either the Euclidean distance or the Manhattan distance. In both cases, moving "closer" to the target on a grid restricted to lines means:

• Moving Right: \( (x, y) \rightarrow (x+1, y) \) (reduces the horizontal gap).
• Moving Up: \( (x, y) \rightarrow (x, y+1) \) (reduces the vertical gap).

Any move in the opposite direction—Left (\( x \) decreases) or Down (\( y \) decreases)—would increase the distance to the top-right corner, violating the condition.

Step-by-Step Reasoning

1. Identify Valid Directions: To ensure the distance to \( (4, 4) \) always decreases, every individual step must be either in the positive \( x \) direction (Right) or the positive \( y \) direction (Up). $$ \text{Allowed moves} \in \{ \text{Right, Up} \} $$
2. Evaluate Option (A): The trajectory starts at \( P \), moves Up, then Right, then Up, then Right, and then appears to move Down or Left. Any segment that goes backwards (Down or Left) is invalid.
3. Evaluate Option (B): The trajectory starts at \( P \), moves Right, then Up, then Right, then Up, then Right, and finally Down. The downward segment increases the distance to the target, so it is invalid.
4. Evaluate Option (C): The trajectory starts at \( P \) and consists only of segments going Up and Right. Since every step in this path moves either vertically up or horizontally right, the distance to the top-right corner strictly decreases at every step. This is a valid trajectory.
5. Evaluate Option (D): The trajectory forms a loop starting at \( P \). It moves Right, Up, Left, and Down. The Left and Down movements increase the distance, so this is invalid.

Final Answer

Based on the requirement that the distance to the destination must strictly decrease, the ant can only move in directions that take it closer to the top-right corner (Up and Right). Only option (C) follows this rule throughout its path.