a) Let $C$ be an $n$-point configuration in the closed unit square $[0, 1] \times [0, 1]$, let $m = \lfloor\sqrt{n}\rfloor$, and consider the snake going horizontally from $0 \times 0$ to $1 \times 0$, then vertically up from $1 \times 0$ to $1 \times 1/m$, then horizontally back from $1 \times 1/m$ to $0 \times 1/m$, vertically up from $0 \times 1/m$ to $0 \times 2/m$, horizontally over to $1 \times 2/m$, and so on and so forth all the way up to $1 \times 1$ or $0 \times 1$, depending on whether $m$ is even or odd. The length of the snake is $m + 1 + m \cdot 1/m = m + 2 \le \sqrt{n} + 2$. Of course, the snake does not necessarily pass through any point in $C$, but it comes within $1/(2m)$ of $C$. Thus, in tracing the snake, visit each point of $C$ by darting out, if necessary, to the nearest points in $C$ abreast within $1/(2m)$, and then dart back. This increases the length by at most $n \cdot 2 \cdot 1/(2m) = n/m < \sqrt{n} + 2$, so the length of the visiting path is certainly less than $2\sqrt{n} + 4$.

b) Let again $m = \lfloor\sqrt{n}\rfloor$, and consider an $n$-point subconfiguration $C$ of the lattice $\{i/m \times j/m: i, j = 0, 1, \dots, m\}$. Since any two distinct points in the lattice are at least $1/m$ distance apart, the length of a path through all of $C$ is at least $(n-1) \cdot 1/m \ge (n-1)/\sqrt{n} \ge \sqrt{n} - 1$.