I have posed the following question on math.stackexchange.com but have not received an answer. So I would like to seek experts' opinion here.

Consider the set of all binary sequence of length $n+1$, $B=\big\{(b_i)_{i=0}^n\,\big| b_i\in\{0,1\}, \forall i\big\}$. Construct a function $f: \{0,\cdots,n\}\times \{0,1\}\to \{0,\cdots, n\}$, such that $\forall (b_i)_{i=0}^n\in B,\,\exists i \colon f(i,b_i)=\sum_{j\ne i}b_j$.

What is a systematic way to construct this function?

Putting it more colloquially, we assign $n+1$ persons one-to-one to all the digits of an arbitrary binary sequence of length $n+1$. Each person can see but the digit assigned to him. Devise a strategy so that at least one person guesses correctly the sum of the remaining digits.