MathDB

Mazo performs the following operation on triplets of non-negative integers: If at least one of them is positive, it chooses one positive number, decreases it by one, and replaces the digits in the units place with the other two numbers. It starts with the triple

x

y

z

. Find a triple of positive integers

x

y

z

such that

xy + yz + zx = 1000

(*) and the number of operations that Mazo can subsequently perform with the triple

x, y, z

is
(a) maximal (i.e. there is no triple of positive integers satisfying (*) that would allow him to do more operations);
(b) minimal (i.e. every triple of positive integers satisfying (*) allows him to perform at least so many operations).

Problem Statement