Tool to calculate all paths on a lattice graphe (square grid graph). A path is a series of directions (north, south, east, west) to connect two points on a grid.

dCode is free and its tools are a valuable help in games, puzzles and problems to solve every day! You have a problem, an idea for a project, a specific need and dCode can not (yet) help you? You need custom development? Contact-me!

The information on this page is for a square grid and is not valid on triangular grids (or other non square lattice graphs).

Tool to calculate all paths on a lattice graphe (square grid graph). A path is a series of directions (north, south, east, west) to connect two points on a grid.

Answers to Questions

How to count paths on a lattice graph?

The calculation of the number of paths (of length \(a + b\)) on a grid of size (a x b) (limited to a north-south direction and a west-east direction) uses combinatorics tools such as the binomial coefficient \( \binom{a+b}{a} \)

The north direction N consists of moving up one unit along the ordinate (0,1).

The east direction E consists of moving one unit to the right along the abscissa (1,0).

Example: To go from the point \( (0, 0) \) to the point \( (2, 2) \) (which corresponds to a 2x2 grid) using only north and east. (N,N,E,E), (N,E,N,E), (N,E,E,N), (E,N,E,N), (E,N,N,E), (E,E,N,N) so 6 paths and is computed \( \binom{4}{2} = 6 \)

What is a lattice graph?

A grid graph is the name given to a bounded grid (with borders).

Example: N,N,N,E has 4 distinct permutations: (N,N,N,E) (N,N,E,N) (E,N,N,N) (N,E,N,N)

Ask a new question

Source code

dCode retains ownership of the source code of the script Lattice Path online. Except explicit open source licence (indicated Creative Commons / free), any algorithm, applet, snippet, software (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt, encrypt, decipher, cipher, decode, code, translate) written in any informatic langauge (PHP, Java, C#, Python, Javascript, Matlab, etc.) which dCode owns rights will not be given for free. To download the online Lattice Path script for offline use on PC, iPhone or Android, ask for price quote on contact page !