# Guessing the number of other $1$'s in a binary sequence

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.

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 $$k\in \{1,...,n\}$$.
• +1 and accepted. Darn, it is so easy when the answer is revealed! I had person $k$ guess $k$ $\forall k$. But then the strategy is one short of covering all $n+2$ cases. Having person $0$ take care of the remaining two cases is the key and is brilliant. – Hans Sep 23 '19 at 3:06