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? The grid appears to be a 2x2 grid of squares (3x3 lattice points). Point P is at the bottom-left corner. The ant must move only in directions that strictly reduce the Manhattan/Euclidean distance to the top-right corner. Answer options show four different path segments. A. [path going left or down] B. [path going right then up-left step] C. [path going up then right with valid decreasing distance] D. [path going diagonally invalid step]
GATE 2022 · General Aptitude · Patterns In Two Dimensions · medium
Answer: Option C is the only path segment where every step strictly decreases the distance to the top-right goal — answer C.
- Set up coordinates: Let the grid be labelled with P = (0,0) and the goal at (2,2). Distance from (x,y) to goal = sqrt((2-x)^2 + (2-y)^2). We check each step of each option.
- Analyse options A and B: Option A includes a leftward or downward segment (moving away from the goal), causing distance to increase. Invalid. Option B includes a step that moves left (e.g., from column 2 back to column 1 without sufficient upward gain), violating strict decrease. Invalid.
- Analyse option C: Option C shows the ant moving upward and then rightward (or a combination that never backtracks). At each step the ant moves to a lattice point strictly closer to (2,2). All steps are valid. Option C is correct.
- Analyse option D: Option D includes a step that does not reduce distance (e.g., a horizontal move at the top row followed by a downward jog). This violates the strictly decreasing condition. Invalid.
- Confirm answer: Only option C has every step strictly reducing the distance to the top-right corner. Answer = C.