>> )> ? ? ? ? sum of digits of 3^1000 ~ 4.5*477 = 2146.5

>> )> which is astonishingly near to the true answer 2142.

>> ) The last digit is 1 (Powers of 3 end in 3, 9, 7, 1, 3, 9, 7, 1 etc.);
>> ) taking that into account would have made it 2143.

>> Taking into account that it has to be a multiple of 9,
>> you can round that to 2142. But that's not proof that it is 2142.
>> Printing the first 30 digit sums doesn't show any pattern either.

> Not a mathematical proof, but you can use statistical arguments
> to show how likely it is to be correct.

> If you can assume that the digits are uncorrelated, then the
> standard deviation of the sum is about 4.5/sqrt(477),
> for about 0.2 That makes it extrememly unlikely to be
> off by more than one.

> The lowest and highest digits aren't so uncorrelated, though.

> It would take more math than I know to show the digit correlation
> is low enough, though.