Lattice Paths of Bounded Length

Let SS be the set of {(1,1),(1,1),(1,1)}\{(1,1), (1,-1), (-1,1)\}-lattice path which begin at (1,1)(1,1), do not use the same vertex twice, and never touch either the xx-axis or the yy-axis. Determine the largest value of nn such that every path in SS which ends at (n,n)(n,n) has length at most 5000050000.

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 (x1,y1)(x_1,y_1) to (x2,y2)(x_2,y_2) is (x2x1,y2y1)(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={(xi,yi)}i=1kS = \{(x_i,y_i)\}_{i=1}^{k}, an SS-lattice path is a lattice path where every step has size which is a member of SS.


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