A lattice path is a path that moves only horizontally or vertically on a grid. The path follows the grid lines, meaning it can only move to adjacent cells that share an edge. This type of path is commonly used in combinatorial problems, where the goal is to find the number of ways to reach a certain point on the grid.

One example of a lattice path is the movement of a knight on a chessboard. The knight can move in an “L” shape, meaning it can move two squares horizontally and one square vertically, or two squares vertically and one square horizontally. This movement pattern is constrained to the grid of the chessboard, making it a lattice path. The number of possible moves the knight can make from a certain position can be determined using combinatorial techniques.

Another example of a lattice path is the movement of a robot on a grid of streets. The robot can only move north, south, east, or west on the grid, following the lines of the streets. This movement is also constrained to the grid, making it a lattice path. The number of possible paths the robot can take from one point on the grid to another can be determined using combinatorial techniques.

In both examples, the movement is constrained to the grid, following only horizontal and vertical paths. This type of movement is useful for problems involving counting the number of possible paths or for problems involving optimization, such as finding the shortest path from one point to another.

Lattice paths can also be extended to three-dimensional grids, such as a cube. In this case, the path can move in the horizontal and vertical directions, as well as up and down along the z-axis. The number of possible paths on a three-dimensional grid can also be determined using combinatorial techniques.

Overall, lattice paths are useful for problems involving movement on a grid, allowing for the counting and optimization of possible paths.