Do I Know a Number Other Than 4?

Let N $= 4444^{4444}$ . The maximum number of digits in N is less than $4444\times4 = 17776$ . Hence, the maximum possible value of A is $17776\times9 = 159984$ .

Similarly, the maximum possible value of B is $45$ & the sum of the digits of B $\leq 12$ .

Observe, that N $≡$ A $≡$ B $(mod 9)$ & $4444$ $≡$ $7$ $(mod 9)$ .

Since $7^{3}$ $≡$ $1$ $(mod 9)$ , we get, $4444^{4444}≡7^{4444}≡7$ $(mod 9)$ as $4444≡1 (mod3)$ .

Hence, the sum of the digits of B is $\boxed{7}$ .

Python:

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def digit_sum(n):
    total = 0
    for digit in str(n):
        total += int(digit)
    return total
a = digit_sum(4444**4444)
b = digit_sum(a)
print "Answer:", digit_sum(b)