Question 133
$n$ is a positive integer. Consider all $2^n$ sequences of numbers $0$ and $1$ with length $n$. At first, some of these numbers are marked. Two sequences are called *neighbors* if they have the same value in all their $n$ digits except for one. In each step, a non-marked sequence that has at least two marked neighbors is also marked. The goal is to have all the sequences to be marked at the end. Find the least number of initially marked sequences such that the goal is achievable.