Very interesting problem...

[musings comment=They turn into good stuff, I promise]
I think that taking the problem in averages would help here.

There are 24 different numbers, each of 4 digits, though only 4 actually different single digit numbers per digit of each of the 4-digit numbers. Digit. For the sum of these 24 different numbers, we can take each digit of the number, average it, multiply it by 10n depending on its placement in the original number, and add them all together. Aka, here:

Let's choose three different single-digit numbers, a; b; c; and d.

In the 26 possible combinations of those 4 digits, each one will repeat at each decimal place 24/4 or 6 times. a*1000 + b*100 + c*10 + d*1, b*1000 + a*100 + c*10 + d*1, etc.

This can then be simplified to 6*a*1000 + 6*b*1000 + 6*c*1000 + 6*d*1000 + 6*a*100 + 6*b*100 + ...

This can then, again, of course, obviously, and to avoid putting us in a coma (sorry for the bad pun ) be simplified to 6*([a+b+c+d]*1000 + [a+b+c+d]*100 + [a+b+c+d]*10 + [a+b+c+d]*1).

This can then, again, yet of course, be simplified to [a+b+c+d]*(6666), meaning that the original number was divisible by 6666 (the sum of the 26, before another number was subtracted from it).


So, what was the original sum? Logically, it has to be greater than 122,000. Also, the very smallest number from four different single-digits stuck together subtracted from it would be 1234, and the very largest one would be 9876, So, we're looking between 122,000+1234 and 122,000+9876, inclusive. Ok, I really hope that there is only one multiple of 6666 in that range: yay, there's only one. Who'da thunk that one: 126,654, or 19*6666.

So, [a+b+c+d] = 19. And what digits of what we subtracted do we know? 126,654 - 122**0 = 00[3 or 4]**4. The [3 or 4] comes from either carrying the 1 from the hundred's place, or not.

So then, this simplifies to 3**4, because each of these 4 digits have to be different. 3+4 = 7, 3+4+*+*=19, so *+*=12. What single-digit numbers add up to 19, and make you carry the hundred's place in that particular subtraction? Well, maybe 7 and 5: let's try. 126,654 - 3754 = 122900. Yay! It worked. Are there any others? 4 and 8, but 4 has been used. 3 and 9, but 9 has been used. 2 and 10, but 10 is a two-digit number. 1 and 11, but that just don't work none.

So, the number subtracted was 3754. Hope that helps, it was fun
[/musings]