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Discrete Mathematics Level pending

Let \(S\) be the set of \( \{(1,1),\) \((1,-1)\), \((-1,1)\), \((1,0)\), \((0,1)\}\)-lattice paths which begin at \((1,1)\), do not use the same vertex twice, and never touch either the \(x\)-axis or the \(y\)-axis. Let \(S_{x,y}\) be the set of paths in \(S\) which end at \((x,y).\) For how many ordered pairs \((x,y) \) subject to \(1 \leq x,y \leq 31\), is \(|S_{x,y}|\) a multiple of \(3\)?

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 \(S\).


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