Find a four-digit number with four different digits, that is equal to the number formed by its digits in descending order minus the number formed by its digits in ascending order.

Looking at the problem, it can be proven that the answer's first number + fourth number = 10, and its second and third number = 8.

The difference between the greatest and least number in the answer has to be the first number in the answer.

9 and 8 can't be in the middle, or you would get -1 and 0 as the other number, which are illegal.

Neither x nor y can be the largest or the smallest, since that would end up with a 0 (as largest or smallest) So x and y have to be between the other numbers. This means that the numbers in the middle have to be the largest and the smallest, and this means that the first number is the difference of the middle ones. Since the first and last number equal 10, you can figure out what numbers could work.

044T The first number is 0, and that is excluded. (Also, T isn't a valid number in base 10)

2358 or 2538 The first number is 5-3=2, so the last number is 10-2=8, But this isn't allowed, since the first and last numbers have to be between the middle numbers' values.

4266 or 462 The first number is 6-4=2, and the last number is 10-4=6. But this isn't allowed, since the first and last numbers have to be between the middle numbers' values.

6174 or 6714 The first number is 7-1=6 and the last number is 10-6=4. This seems to work so far, so to check it, you get 7641-1467=6174, and you see that the first one works.

So the answer must be 6174.

Posted by Gamer on 2003-04-26 11:26:20