Foot Pursuit on the Grid
At 21:40, Officer Hale is on foot at the corner of 5th Street and Main Street, facing north on Main. Main Street runs north–south; 5th Street runs east–west; every block is the same length and the intersections are square. Radio traffic directs her after a suspect: north two blocks, right three blocks, right one block, left two blocks, right one block. She stops at that point. Meanwhile, Sergeant Ochoa is standing at 5th and Main and wants to move to Hale's final position by the shortest street route.
1. State Hale's final position relative to the start, and the direction she is facing.
Read the worked answer
Track the turns one at a time. Let the start at 5th and Main be (0,0), east positive, north positive.
| Move | Facing before | Position after |
|---|---|---|
| North 2 | north | (0, 2) |
| Right 3 | east | (3, 2) |
| Right 1 | south | (3, 1) |
| Left 2 | east | (5, 1) |
| Right 1 | south | (5, 0) |
Final position: five blocks east of Main Street, on the same east–west street as the start. Facing: south.
Why the turn directions matter: turning right from north gives east; right from east gives south; left from south gives east; right from east gives south. A single misread turn - especially the left after two rights - moves the answer by two blocks.
2. What is the shortest street route for Ochoa from 5th and Main to Hale's final position, and how many blocks is it?
Read the worked answer
Distance: five blocks. Hale's final position is (5, 0) - five blocks due east of Ochoa, on the same street.
Route: proceed east on 5th Street for five blocks, directly along the same east–west line. No turns are needed, because the net north–south displacement is zero.
Why no alternate is shorter: on a square grid with equal blocks, any route mixing north–south travel adds blocks without reducing east–west travel. The minimum is the Manhattan distance, |5 − 0| + |0 − 0| = 5.
Useful check: Hale's northward and southward moves cancel (2 north, 1 south, 1 south), which is exactly why the final position sits back on the starting street line.
3. If Hale had turned right instead of left at the third turn, and had then made the same final right turn, where would she be and which way would she face?
Read the worked answer
Recompute only from the changed turn. Up to the second turn she is at (3, 1) facing south.
| Move | Facing before | Position after |
|---|---|---|
| Right 2 (instead of left 2) | west | (1, 1) |
| Right 1 (same final right) | north | (1, 2) |
Final position: one block east and two blocks north of the start. Facing: north.
Why the facing flips: facing south, a right turn points west rather than east; facing west, the final right turn points north. The position changes by four blocks east–west compared with the original path, so this is not a small error to absorb by guessing.
Shortest route for Ochoa in this variant: one block east on 5th, then two blocks north - three blocks total.