Grids and Movement

Robots on a grid follow commands like F, L and R. We track the cell and the facing, find the fewest moves and count shortest routes.

Key ideas

Cells are named (column, row), with (1, 1) at the bottom left; the column is always written first.
F moves one cell forward in the direction the robot faces; L and R turn it 90° and it stays in the same cell.
Facing North the row goes up; East, the column goes up; South, the row goes down; West, the column goes down.
With no walls, the fewest moves between two cells is the steps across plus the steps up or down.
In a maze, count outwards from the start one ring of squares at a time until you reach the goal.
For routes moving only right or up, each corner’s count is the corner to its left plus the corner below.

Remember

Turning right from North	North → East → South → West → North
Four right turns	back to the starting direction
Fewest moves from (2, 3) to (6, 1)	4 across + 2 down = 6
Back to the start	difference of East and West steps + difference of North and South steps
Routes on a 2 × 2 street grid	6

Watch out

Writing (row, column) instead of (column, row).
Moving the robot when it only turns: L and R do not change the cell.
Turning the wrong way: a right turn from North faces East, not West.

Worked example

(a) On a street grid you walk from corner (0, 0) to corner (3, 4), moving only right (east) or up (north), one block at a time. How many different shortest routes are there?
(b) On a grid of streets, a delivery robot goes 7 steps West, then 6 steps North, then 7 steps South, then 1 step South, then 1 step West. It can only move North, South, East or West. What is the fewest number of steps to get back to the start?

Working: (a) write at each corner the number of ways to reach it: the corner to its left plus the corner below; (b) it ends 8 West and 2 South of the start

Answer: (a) 35; (b) 10