I listed a few numbers at random: 5, 4, 3, 8. 8543-3458=5085; 8550-0558=7992; 9972-2799=7 173; 773 1- 1377=6354; 6543-3456=3087; 8730-0378=8352; 8532-2358=6 174
This is a mathematical black hole problem.
Take any four digits, as long as they are not all the same, and arrange them in descending order to form the largest number as the minuend; According to the ascending order of numbers, take the smallest number for subtraction, and the difference will be 6174; If it is not 6 174, subtract it according to the above method and get 6 174 in no more than 7 steps.
If you take the four digits of 5462, do the following operation according to the above method:
6542-2456=4086 8640-0468=8 172
872 1- 1278=7443 7443-3447=3996
9963-3699=6264 6642-2466=4 176
764 1- 1467=6 174
So, what is the scientific basis for the result of 6 174?
Let m be a four-digit number and all four digits are different, and arrange the numbers of m in descending order.
Marked as m (minus);
Then the numbers in m are arranged in ascending order, marked as m increasing, and the difference m (minus) -M (plus) =D 1. From m to D 1, we regard it as a transformation, and the transformation from m to D 1 is marked as: T (m) = D65438+.
Note: T(D 1)= D2.
Similarly, D2 can be converted into D3; D3 is converted into D4, that is, t (D2) = D3 and t (D3) = D4. ...
Now we have to prove that if we repeat the transformation at most seven times, we will get D7=6 174.
Certificate: There are 65,438+004 = 65,438+00,000 four digits, among which, except the four digits are all the same, the other 65,438+004-65,438+00 = 9990 digits are not all the same. We first prove that the transformation T only transforms these 9990 numbers into 54 different four-digit numbers.
Let a, b, c and d be numbers of m, and let:
a≥b≥c≥d
Because not all of them are equal, the equal signs in the above formula cannot be established at the same time. We calculate T(M)
M (minus) =1000a+100b+10c+d
M (increase) =1000d+100c+10b+a.
T(M)= D 1= M (minus) -M (plus) =1000 (a-d)+100 (b-c)+10 (c-b)+d-.
We notice that T(M) only depends on (a-d) and (b-c), because a, b, c and d are not all equal, so it can be deduced from a≥b≥c≥d; A-d > 0 a-d>0 b-c ≥ 0。
In addition, B and C are between A and D, so a-d≥b-c, that is to say, a-d can take 9 values of 1, 2, …, 9. If a certain value of this set is taken, b-c can only take a value less than N at most.
For example, a-d= 1, then b-c can only be selected from 0 and 1. In this case, T(M) can only take the value:
999×( 1)+90×(0)=0999
999×( 1)+90×( 1)= 1089
Similarly, if a-d=2, T(M) can only take three values corresponding to b-c=0, 1, 2. When a-d= 1, a-d=2, …, a-d=9, we add up the possible values of b-c.
This is the number of possible values of T(M). Among the 54 possible values, some are values with the same number but different digits. Convert these values into the same value in T(M) (these two numbers are equivalent in mathematics) and eliminate the equivalence factor. Of the 54 possible values of T(M), only 30 are not equivalent. They are:
9990,998 1,9972,9963,9954,98 10,97 1 1,962 1,953 1,944 1,8820,8730,872 1,8640,8622,8550,
8532,8442,773 1,764 1,7632,755 1,7533,7443,6642,6552,6543,5553,5544.
For these 30 numbers, use the above rules to change them into the difference between the maximum number and the minimum number, and the number 6 174 will appear in at most 6 steps.