How many steps do I need?

Question

The start point is $(0,0,0)$. The "step vector" is $(2,3,6)$. I can variate this, so it can be $(3,6,2)$ and so on, and it can be negative too, like $(6,2-,3)$. I can't step over $(99,99,19)$ and I can't step under $(0,0,0)$. What is the minimum step to reach the finish $(99,99,19)$ point, if I can? And with other step vector, $(2,3,5)$ and $(2,3,4)$? I tried write a solver program in python but it doesn't work yet.

Answer 1

If we use the Manhattan metric in the first two coordinates and think of having your steps of $2$ in the third you need to get $198$ from the origin with steps of length $9$. It takes $22$ steps for that. Using $2$ in the third coordinate all the time will not work because $19$ is odd. We imagine using one $3$ in the third coordinate, which makes the total distance we can get in the first two coordinates $197$. We can add one with one more step by swapping two more $3$s into the third axis and negating a $3$. We can then write $$99=10\cdot 6 + 13 \cdot 3\\99=13\cdot 6+6\cdot 3+1\cdot(-3)+3\cdot 2\\ 19=2\cdot 3 + 1\cdot (-3)+14\cdot 2 +6 \cdot (-2)$$ Note that the coefficients on each line sum to $23$. You need an arrangement of steps that satisfies your constraint that the third coordinate does not exceed $19$