Find the number of ordered triples $$(x, y, z)$$ of positive integers such that $$1 \le x, y, z \le 8$$ and $$|x - y| + |y - z| + |z - x| = 8$$.

If $$m, t, M$$ are the smallest, middle and largest of the three values, the sum of the three distances equals $$2(M - m)$$, so $$M - m = 4$$ and $$(m, M)$$ is one of $$(1, 5), (2, 6), (3, 7), (4, 8)$$. For each pair, a strictly intermediate middle value gives $$3 \times 3! = 18$$ triples and a middle value equal to an endpoint gives $$2 \times 3 = 6$$ triples, so 24 in all. The total is $$4 \times 24 = 96$$.