[ned_budge]

I assume you mean the single digit is always equal to 9.

When you add the digits of a number repeatedly until there is a single digit, it is call the digital root of that number. The digital root of a number is the value of the remainder when you divide a number by 9. Assume you have 2 numbers A and B. Subtract B from A and get the result C. The digital root of A minus the digital root of B should equal the digital root of C.

By scrambling the same digits, what you have is two numbers with the same digital root. By subtracting the numbers the result will be a number with the digital root of zero. There is only two single digit numbers that have a digital root of zero, 0 and 9. By scrambling the digits of the original number will result in a number greater than 0 and of course if the number is greater than zero then sum of the digits will be greater than zero, which means the result will alway be 9.

You may remember a trick that was taught in grade school to tell if a number is divisible by 9; add the digits together and if the result is divisible by 9 then so is the number. Same trick works for numbers divisible by 3. Same principle.

[Publius6961]

Thank you. Best explanation I've received yet.
I have to look more into this digital root business; if I ever learned it, I sure don't remember.

Is it easier than the first derivative of the transcendental functions?