To apply Kaprekar's routine to the 6-digit number 309612 , we follow a process of rearranging the digits to find the maximum and mini...
To apply Kaprekar's routine to the 6-digit number 309612, we follow a process of rearranging the digits to find the maximum and minimum possible numbers, subtracting them, and repeating the process until we reach a constant or a cycle.
Here is the step-by-step calculation:
The Routine
Step 1:
Digits: 3, 0, 9, 6, 1, 2
Descending (Max): 963,210
Ascending (Min): 012,369
Calculation: $963,210 - 012,369 = \mathbf{950,841}$
Step 2:
Digits: 9, 5, 0, 8, 4, 1
Descending (Max): 985,410
Ascending (Min): 014,589
Calculation: $985,410 - 014,589 = \mathbf{970,821}$
Step 3:
Digits: 9, 7, 0, 8, 2, 1
Descending (Max): 987,210
Ascending (Min): 012,789
Calculation: $987,210 - 012,789 = \mathbf{974,421}$
Step 4:
Digits: 9, 7, 4, 4, 2, 1
Descending (Max): 974,421
Ascending (Min): 124,479
Calculation: $974,421 - 124,479 = \mathbf{849,942}$
Step 5:
Digits: 8, 4, 9, 9, 4, 2
Descending (Max): 998,442
Ascending (Min): 244,899
Calculation: $998,442 - 244,899 = \mathbf{753,543}$
Step 6:
Digits: 7, 5, 3, 5, 4, 3
Descending (Max): 755,433
Ascending (Min): 334,557
Calculation: $755,433 - 334,557 = \mathbf{420,876}$
Observations
Unlike 4-digit numbers (which always lead to 6174), 6-digit numbers do not have a single constant. Instead, they usually fall into one of two "loops." Based on the result of Step 6 (420,876), your number has entered a common 7-step constant loop:
420,876 $\rightarrow$ 876,420 - 024,678 = 851,742
851,742 $\rightarrow$ 875,421 - 124,578 = 750,843
750,843 $\rightarrow$ 875,430 - 034,578 = 840,852
840,852 $\rightarrow$ 885,420 - 024,588 = 860,832
860,832 $\rightarrow$ 886,320 - 023,688 = 862,632
862,632 $\rightarrow$ 866,322 - 223,668 = 642,654
642,654 $\rightarrow$ 665,442 - 244,566 = 420,876 (Back to the start of the loop)
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