The first useful observation is that individual binary digits of the friendship value are mutually independent, so they can be considered separately. If the ~i^\text{th}~ digit of a result is equal to ~1~, a value of ~2^i~ is added to the friendship value, otherwise ~0~ is added.

A digit of the result is equal to ~1~ only if the corresponding digits in extraterrestrial names differ. Since we are computing the total friendship value, we can count the number of appearances of ~2^i~ (digit ~1~ in position ~i~) for each ~i~.

Let us denote by ~K_i~ the number of extraterrestrials who have the digit ~1~ in position ~i~ in binary name notation. Then we add

$$\displaystyle (K_i \times (N-K_i)) \times 2^i$$

to the sum of friendships, since that is exactly the number of pairs with differing digits in position ~i~, multiplied by the weight of that digit position.