I'm not sure who is still paying attention to this thread, but here goes anyway.
First, in the official-looking linked version, it only has to be 1000 factorial, not 10000 factorial. Also, when this problem was reused in another programming contest, the time limit was 3 seconds, not 1 second. This makes a huge difference in how hard you have to work to get a fast enough solution.
Second, for the actual parameters of the contest, Peter's solution is sound, but with one extra twist you can speed it up by a factor of 5 with 32-bit architecture. (Or even a factor of 6 if only 1000! is desired.) Namely, instead of working with individual digits, implement multiplication in base 100000. Then at the end, total the digits within each super-digit. I don't know how good a computer you were allowed in the contest, but I have a desktop at home that is roughly as old as the contest. The following sample code takes 16 milliseconds for 1000! and 2.15 seconds for 10000! The code also ignores trailing 0s as they show up, but that only saves about 7% of the work.
#include <stdio.h>
int main() {
unsigned int dig[10000], first=0, last=0, carry, n, x, sum=0;
dig[0] = 1;
for(n=2; n <= 9999; n++) {
carry = 0;
for(x=first; x <= last; x++) {
carry = dig[x]*n + carry;
dig[x] = carry%100000;
if(x == first && !(carry%100000)) first++;
carry /= 100000; }
if(carry) dig[++last] = carry; }
for(x=first; x <= last; x++)
sum += dig[x]%10 + (dig[x]/10)%10 + (dig[x]/100)%10 + (dig[x]/1000)%10
+ (dig[x]/10000)%10;
printf("Sum: %d
",sum); }
Third, there is an amazing and fairly simple way to speed up the computation by another sizable factor. With modern methods for multiplying large numbers, it does not take quadratic time to compute n!. Instead, you can do it in O-tilde(n) time, where the tilde means that you can throw in logarithmic factors. There is a simple acceleration due to Karatsuba that does not bring the time complexity down to that, but still improves it and could save another factor of 4 or so. In order to use it, you also need to divide the factorial itself into equal sized ranges. You make a recursive algorithm prod(k,n) that multiplies the numbers from k to n by the pseudocode formula
prod(k,n) = prod(k,floor((k+n)/2))*prod(floor((k+n)/2)+1,n)
Then you use Karatsuba to do the big multiplication that results.
Even better than Karatsuba is the Fourier-transform-based Schonhage-Strassen multiplication algorithm. As it happens, both algorithms are part of modern big number libraries. Computing huge factorials quickly could be important for certain pure mathematics applications. I think that Schonhage-Strassen is overkill for a programming contest. Karatsuba is really simple and you could imagine it in an A+ solution to the problem.
Part of the question posed is some speculation that there is a simple number theory trick that changes the contest problem entirely. For instance, if the question were to determine n! mod n+1, then Wilson's theorem says that the answer is -1 when n+1 is prime, and it's a really easy exercise to see that it's 2 when n=3 and otherwise 0 when n+1 is composite. There are variations of this too; for instance n! is also highly predictable mod 2n+1. There are also some connections between congruences and sums of digits. The sum of the digits of x mod 9 is also x mod 9, which is why the sum is 0 mod 9 when x = n! for n >= 6. The alternating sum of the digits of x mod 11 equals x mod 11.
The problem is that if you want the sum of the digits of a large number, not modulo anything, the tricks from number theory run out pretty quickly. Adding up the digits of a number doesn't mesh well with addition and multiplication with carries. It's often difficult to promise that the math does not exist for a fast algorithm, but in this case I don't think that there is any known formula. For instance, I bet that no one knows the sum of the digits of a googol factorial, even though it is just some number with roughly 100 digits.
