// for a = z, b = x + y, there are some solutions if and only if b > a/2, and the amount
//can be (sneakily) calculate with some clever arithmatic to be (b - (a + 1) /2);
private long CountShortestPathsBySplittingB(Pythagorean.Triple triple)
{
return(Math.Max(0, (triple.a - (triple.b + 1) / 2) + 1));
}
// If the Cube lengths are x,y,z then (x+y)^2 + z^2 being the shortest path ==> x <= z, y <= z
// (To see why, map out all three shortest path candidates, assert one to be smaller than the other two,
// and look at what this implies about x, y, and z. Algebra is very straightforward)
// so given a triple of a,b,c, we are only interested in finding x,y,z
// such that WLOG a = x + y, b = z OR a = z, b = x + y where our inequality holds (x, y <= z)
// for a = x + y, b = z, then there are a/2 ways of finding x,y such that x+y = a and x,y < z
private long CountShortestPathsBySplittingA(Pythagorean.Triple triple)
{
return(triple.a / 2);
}
