I have posed the following question on [math.stackexchange.com](https://math.stackexchange.com/q/3365266/64809) 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?




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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. 



Guessing the number of other $1$'s in a binary sequence
Since every person knows his assigned digit, the problem is equivalent to guessing the sum of all digits.


Label the persons with $0,1,...,n$. Let person $0$ guess $0$ when given a $0$, and $n+1$ when given a $1$. Let person $k$ guess $k$ for every $k\in \{1,...,n\}$.


At least one person can correctly guess the sum of all digits. So when they subtract their given digit from the guessed sum, at least one person can correctly guess the sum of the remaining digits. 


