  In this paper, we consider the gathering problem of seven autonomous mobile
robots on triangular grids. The gathering problem requires that, starting from
any connected initial configuration where a subgraph induced by all robot nodes
(nodes where a robot exists) constitutes one connected graph, robots reach a
configuration such that the maximum distance between two robots is minimized.
For the case of seven robots, gathering is achieved when one robot has six
adjacent robot nodes (they form a shape like a hexagon). In this paper, we aim
to clarify the relationship between the capability of robots and the
solvability of gathering on a triangular grid. In particular, we focus on
visibility range of robots. To discuss the solvability of the problem in terms
of the visibility range, we consider strong assumptions except for visibility
range. Concretely, we assume that robots are fully synchronous and they agree
on the direction and orientation of the x-axis, and chirality in the triangular
grid. In this setting, we first consider the weakest assumption about
visibility range, i.e., robots with visibility range 1. In this case, we show
that there exists no collision-free algorithm to solve the gathering problem.
Next, we extend the visibility range to 2. In this case, we show that our
algorithm can solve the problem from any connected initial configuration. Thus,
the proposed algorithm is optimal in terms of visibility range.
