# Zig-zag through the maize

**Discrete Mathematics**Level 4

Let \(S\) be the set of \(\{(1,1), (1,-1), (-1,1)\}\)-lattice path which begin at \((1,1)\), do not use the same vertex twice, and never touch either the \(x\)-axis or the \(y\)-axis. How many paths in \(S\) end at the point \((4,8)\)?

**Details and assumptions**

A **lattice path** is a path in the Cartesian plane between points with integer coordinates.

A **step** in a lattice path is a single move from one point with integer coordinates to another.

The **size** of the step from \((x_1,y_1)\) to \((x_2,y_2)\) is \((x_2-x_1,y_2-y_1)\).

The **length** of a lattice path is the number of steps in the path.

For a set \(S = \{(x_i,y_i)\}_{i=1}^{k}\), an \(S\)-lattice path is a lattice path where every step has size which is a member of \(S\).