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Problem of the Week
Problem E
The Shortest Path

On the Cartesian plane, we draw grid lines at integer points along the \(x\) and \(y\) axes. We can then draw paths along these grid lines between any two points with integer coordinates. The graph below shows two paths along these grid lines from \(O(0,0)\) to \(P(6,-4)\). One path has length \(10\) and the other has length \(20\).

One path starts at O, then goes one unit down, then three units to the right, then two down, then three to the right, then one down arriving at P. The other path starts at O, then goes two units to the right, then three up, then six to the right, then seven down, then two to the left arriving at P.

There are many different paths along the grid lines from \(O\) to \(P\), but the smallest possible length of such a path is \(10\). Let’s call this smallest possible length the path distance from \(O\) to \(P\).

Determine the number of points with integer coordinates for which the path distance from \(O\) to that point is \(10\).