A bug travels in the coordinate plane moving only along the lines that are parallel to the $$x$$ axis or $$y$$ axis. Let $$A = (-3, 2)$$ and $$B(3, -2)$$. Consider all possible paths of the bug from $$A$$ to $$B$$ of length at most 14. How many points with integer coordinates lie on at least one of these paths?

A lattice point $$P$$ lies on such a path exactly when the taxicab distances satisfy $$d(A,P) + d(P,B) \le 14$$, and this total always has the same parity as $$d(A,B) = 10$$. Writing the condition as $$f(x) + g(y) \le 14$$ where $$f(x) = |x+3| + |x-3|$$ and $$g(y) = |y-2| + |y+2|$$, one has $$f(x) = 6$$ for $$|x| \le 3$$ and $$2|x|$$ otherwise, with a similar description for $$g$$. Counting the lattice points satisfying this gives 87.