Equivalence between connecting $n\times n$ grid and the gossip problem In this thread, the problem about connecting a $n\times n$ grid without lifting your pen was discussed.
A user mentioned the relation between this problem and the gossip problem.

Apparently this is equivalent to the gossip problem. (see A058992.) – Vepir

I find such equivalence not-so-obvious. Wonder if anyone can provide some insights (or complete proof, if possible) about the equivalence between the connect $n\times n$ grid problem and the gossip problem.
P/s. It would have been easier for me if I have more than 50 reputations to ask the user directly about the relationship instead of posting this question.
 A: I think there is nothing deep going on here. Let us examine the original comment on https://oeis.org/A058992 which inspired Vepir to make their comment.

The sequence (for n>=1) refers to the famous "nine dots puzzle" as well. It represents the minimum number of straight lines that you need to fit the centers of n^2 dots (without lifting the pencil from the paper). - Marco Ripà, Apr 01

Let $t_n$ the minimum number of calls in the solution to the gossip problem, and let $g_n$ be the minimal number of segments in a path through all points in a $n\times n$ grid array. For all we know, Marco Ripà's is just referring to the fact that
$$
g_n=t_{n+1} \qquad \text{for all }n\ge 0
$$
since he is just saying that the $(t_n)_{n\ge 1}$ sequence "represents" the solution to the dot problem. This is of course a pertinent comment to make on an OEIS entry; there is no need to have separate sequences for the two problems when they differ only by a shift.
There are two possibilities here; either there is some hidden isomorphism which shows how the gossip problem is equivalent to the dots problem, or the two problems coincidentally have the same answer. I think coincidence is more likely. Certainly, there are more interesting math problems with simple formulas as answers then there are simple formulas, so there are bound to be many such coincidences.
A: happy to answer this question. The connection between the $n \times n$ dots problem and the gossip problem is that they both points to the OEIS sequence $A058992$, but there is something interesting here, since I have just released a preprint (currently under review in an international journal), which shows that the shortest polygonal chains covering those $n \times n$ grids are not necessarily only covering trails (as proved by Keszegh in 2013), but also covering paths and even covering circuits (!) at least $50\%$ of times. Here is the preprint I am talking about: https://arxiv.org/abs/2207.08708
Moreover, I have published a trilogy of papers extending this kind of problems to $k$-dimesional grids, solving the general $3 \times 3  \dots \times 3$ case, by constructively proving that the minimum-link covering trail has a link-length of (exactly) $\frac{3^k -1}{2}, \forall k \in \mathbb{Z}^{+}$ (see SOLVING THE 106 YEARS OLD 3^k POINTS PROBLEM WITH THE CLOCKWISE-ALGORITHM).
A: The source of the information is that OEIS sequence A058992 gives both the solution to the gossip problem and to the $n\times n$ grid problem.
This is what we expect to happen if the two problems have the same numerical solution, whether or not they are related. The OEIS is not going to have two different entries for the same sequence. So all interpretations of a sequence are going to get the same page.
In some cases, it would be a huge coincidence if two different, unrelated problems produced the same sequence. For example, if the answer to a problem were given by the Catalan numbers (A000108), then I would expect that it is somehow equivalent to the other combinatorial interpretations of the Catalan numbers.
In this case, a coincidence is not so unreasonable, because sequence A058992 is equal to $2n-4$ for all $n \ge 4$, with only a few unusual initial terms. It would not be so surprising if two unrelated problems had this formula.
It also does not seem like there are obvious connections. I am tempted to imagine that the $n \times n$ grid represents the information shared in the gossip problem so far: maybe the $n$ rows represent the $n$ people and the $n$ columns represent the $n$ pieces of information. However, this does not help, because the lines that cover the $n \times n$ grid don't appear to correspond to phone calls in the gossip problem.
I am tentatively guessing that there is no connection, but of course "I did not think of anything clever" is not a proof.
