<P> so any number over 1000 gets smaller under this process and in particular becomes a number with strictly fewer digits . Once we are under 1000, the number for which the sum of squares of digits is largest is 999, and the result is 3 times 81, that is, 243 . </P> <Ul> <Li> In the range 100 to 243, the number 199 produces the largest next value, of 163 . </Li> <Li> In the range 100 to 163, the number 159 produces the largest next value, of 107 . </Li> <Li> In the range 100 to 107, the number 107 produces the largest next value, of 50 . </Li> </Ul> <Li> In the range 100 to 243, the number 199 produces the largest next value, of 163 . </Li> <Li> In the range 100 to 163, the number 159 produces the largest next value, of 107 . </Li>

The difference of two numbers is the same as their product and the sum of their reciprocals is 9