Loop dLoop
Make up a 3digit number whose digits are all different. Using these digits, rearrange them to make the largest and smallest possible 3digit numbers, M and m respectively. Work out their difference (that is, Mm) to get another 3digit number...and repeat the procedure.
For example, starting with 312, we find M=321 and m=123, and have:
321  123 = 198
981  189 = 792
972  279 = 693
963  369 = 594
954  459 = 495
954  459 = 495
A loop has been created...
Try some other numbers...in fact, many other 3digit numbers. What happens? Can you develop arguments to support your conjecture(s)?
Extensions: What happens if you started with 2digit numbers? 4digit numbers? 5digit numbers? ndigit numbers? That is, is there a generalizable pattern that can be justified?
Hint: Look for key numbers that keep appearing as you try 3digit numbers...and ask what can you learn from them?
Solution Commentary: The words "the solution" do not fit here...Rather, your discoveries are a solution. Can you justify them?
Also, were you able to extend your ideas to ndigit numbers? Why or why not?
