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## Loop d-Loop

Make up a 3-digit number whose digits are all different. Using these digits, re-arrange them to make the largest and smallest possible 3-digit numbers, M and m respectively. Work out their difference (that is, M-m) to get another 3-digit 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 3-digit numbers. What happens? Can you develop arguments to support your conjecture(s)?

Extensions: What happens if you started with 2-digit numbers? 4-digit numbers? 5-digit numbers? n-digit numbers? That is, is there a generalizable pattern that can be justified?

Hint: Look for key numbers that keep appearing as you try 3-digit 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 n-digit numbers? Why or why not?