Expert: Sherman D. - 4/17/2004

Question
Niko came home from school and told his grandmother that he had learned an amazing trick in math class. They both sat down at the kitchen table and this is what he said and wrote:

“Select 4 different numbers from 0 to 9”               6 7  4  5
“Rearrange the digits to create the largest number possible”            7654
“Use the same four digits to create the smallest number possible”         4567
“Subtract the smaller number from the larger one”         7654 – 4567 = 3087

“Now take the difference and rearrange the digits to make the largest number possible”    8730
“Rearrange the digits to make the smallest number possible”            0378
“Subtract the smaller from the larger”            8730 – 0378 = 8352

“Repeat Again”   8532 – 2358 = 6174               


Niko's grandmother grew more impressed as he showed her with other numbers. Sometimes he could get the magic number 6174, with only one subtraction. With others it took more. He kept trying different combinations, recording how any subtractions it took to reach 6174, and he started to see some patterns.

Question 1:W what were the patterns Niko found?
Question 2: What do you notice about the 4 digits when only one subtraction is needed to reach 6174?
Question 3: What number(s) require(s) the greatest number of subtractions to reach 6174?


Answer
here is what i can tell you

(1000x + 100(x - 1) + 10(x - 2) + (x - 3)) -
(1000(x - 3) + 100(x - 2) + 10(x - 1) + 1000x) = y

where as x is the largest number possible, and the rest is one less than the number before it.

Sorry i couldn't be of much more help. I suck at word problems.