On the left, we have a starting configuration for a game of Hashi, and on the right, its solution:
That is to say, the goal is to make connections (planar, and traveling only in cardinal directions) from each node such that the total number of connections at each node is in accordance with the node's index, and such that the final graph is connected. You can play around with it here. Note: In this version, only 0, 1, or 2 connections are possible between two nodes, but for the purposes of this question, there may be an arbitrary number of connections between two nodes!
Problem statement: For each integer n > 1, does there exist a nontrivial Hashi puzzle (with a unique solution) using only index-n nodes? "Nontrivial" simply means to exclude the configuration consisting of two adjacent nodes, whose unique solution is an n-fold connection between them. For example, for n=2, any (properly embedded) cycle graph is nontrivial and uniquely solvable.