$\textbf{Answer}$

Joe can make one cut at the $3^{rd}$ link. Then he can take a chain of two links on left off the chain and also the chain of three links on the right off. This will separate the original chain into three segments: one with $2$ links, one with $3$ links and the one link what is cut.

With three segments, $1$-link, $2$-link, and $4$-link, at hand,

• On the $1^{st}$ day, Joe gives the assistant the $1$-link
• On the $2^{nd}$ day, Joe gives the assistant the $2$-link and gets back the $1$-link.
• On the $3^{rd}$ day, Joe gives the assistant the $1$-link again
• On the $4^{th}$ day, Joe gives the assistant the $4$-link and takes back both the $1$-link and the $2$-link
• On the $5^{th}$ day, Joe gives the assistant the $1$-link again
• On the $6^{th}$ day, Joe gives the assistant the $2$-link and takes back the $1$-link
• On the $7^{th}$ day, Joe gives the assistant the $1$-link

$\textbf{Analysis}$

Obviously, this solution assumes that the assistant will not spend silver links received during this week and is willing to give them back as exchange. This is an implicit assumption which makes this problem a brain teaser. (Otherwise, Joe has to cut his gold chain into $7$ pieces which makes this problem no longer a brain teaser.) Therefore, if this problem appears in a face-to-face interview, then it will be a good opportunity to show your understanding by asking for clarification (see the introduction chapter).

$\textbf{Note}$

From a technical point of view, $1$, $2$, $4$ form a sequence of powers of $2$: $1=2^0$, $2=2^1$, and $4=2^2$. Such a sequence can construct any number between $1$ and the sum of the elements (i.e. $7$ in this case).