This article is from the Puzzles FAQ, by Chris Cole firstname.lastname@example.org and Matthew Daly email@example.com with numerous contributions by others.
What are the extremal products of three three-digit numbers using digits 1-9?
There is a simple procedure which applies to these types of problems (and
which can be proven with help from the arithmetic-geometric inequality).
For the first part we use the "first large then equal" procedure.
This means that are three numbers will be 7**, 8**, and 9**. Now
the digits 4,5,6 get distributed so as to make our three number as
close to each other as possible, i.e. 76*, 85*, 94*. The same goes
for the remaining three digits, and we get 763, 852, 941.
For the second part we use the "first small then different" procedure.
Our three numbers will be of the form 1**, 2**, 3**. We now place
the three digits so as to make our three numbers as unequal as possible;
this gives 14*, 25*, 36*. Finishing, we get 147, 258, 369.
Now, *prove* that these procedures work for generalizations of this