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Patterns Come in Threes...Or Do They?


Take a 3-digit number whose digits are all different. For example, 321...

From the three digits, create the largest and smallest possible 3-digit numbers, M and m respectively. For example, M = 321 and m = 123...

Find their difference. For example, 321 - 123 = 198...

Repeat the procedure. For example...

321 - 123 = 198
981 - 189 = 792
972 - 279 = 693
963 - 369 = 594
954 - 459 = 495
954 - 459 = 495
.....A loop!

Try some other 3-digit numbers...in fact, many of them. What happens?

Is a loop always reached?

Do the subtractions always return a 3-digit number...or should numbers such as 95 be represented as 095 to continue the procedure?

What happens if your starting number had only 2 unique digits?

What happens if your intial number was 2-digits? 4-digits? 5-digits? n-digits?

Can you generalize or prove any of your observed patterns or claims?

NOTE: This type of problem is clalled a black-hole. If you want to explore more of this type, see Bob Albrecht's FREE 96-page eBook Mathemagical Black Holes. It includes the 495 black hole mentioned in this MathNEXUS episode and many more mathemagical black holes..claimed to be "the largest collection of mathemagical black holes available on the Internet."

 


Hint: As suggested in the problem's statement, try some other 3-digit numbers...in fact, many of them. Observe what happens.

 


Solution Commentary: To focus your search for patterns, relook at the example...

321 - 123 = 198
981 - 189 = 792
972 - 279 = 693
963 - 369 = 594
954 - 459 = 495
954 - 459 = 495
.....A loop!

Look at the results: 198, 792, 693, 594, 495...See any patterns....middle digit? Sum of outside digits?

So, why does this happen. Suppose your original nunmber was ABC where A>B>C. The, in our first subtraction, we have (100A+10B+C)-(100C+10B+A) = 99A-99C? What happens to the B-term? And, what does 99(A-C) tell you? If you are not sure, go back and look at the 3-digit number 954.