examday

/TMUA

Consider the $n$ digit number $B=b_1b_2\ldots b_n$, where each of $b_i$ ($1\le i\le n$) equals either $0$ or $1$.

A student wishes to deduce each digit $b_i$, thereby deducing $B$. They are allowed to query the digit sum of any 3 distinct digits, i.e. $(b_p+b_q+b_r)$ for $p\ne q\ne r$.

Which of the following must be true?

• I. If $n=4$, then there exists a strategy to deduce $B$ by querying 2 three-digit sums.
• II. If $n=3k$ for some integer $k$, then there exists a $B$ where querying $k+1$ digit sums may be sufficient to deduce $B$.
• III. If $n=10$, then there exists a strategy to deduce $B$ by querying 8 three-digit sums.