Country Road is a logic puzzle by Nikoli.
The square grid is divided into "rooms" of various shapes; some of these rooms contain a number. The object is to draw a closed loop in the grid, entering and exiting each room once. A room with a number must have that many squares used by the loop (a room without a number may have any number of its cells used). Lastly, orthogonally adjacent unused squares must not be in different rooms.
Rooms with given numbers of cells are often a good starting point, especially if they have as many cells used as cells in that room.
Rooms that are only adjacent to two other regions must connect to those two other regions. If those two other regions connected, that would form a smaller closed loop than required, so they cannot connect with each other, and so one can mark any common borders of those rooms as "unused".
Marking in impossible places for loop segments can eventually lead to certain cells being unused, and so orthogonally adjacent cells in other regions being used.
As with many loop puzzles, a corollary of the Jordan Curve Theorem (i.e. any two closed loops must intersect an even number of times) can help determine whether a loop segment is used or not.
In all capital-T-shaped rooms, one can infer that at least one of the legs in that capital-T-shaped room will not be used. Thus for any such shaped rooms which is made of only 4 cells (thus one cell per leg), one can infer that at least one of those cells will not be used. Obviously, there could be more than one cell not used, but you can only guarantee the minimum of one cell not being used.
Similarly, for all plus-shaped rooms, one can infer at least two legs in that plus-shaped room are not used. Thus, in a 5 cell variant of this kind of room, you can infer at least two of those cells (thus one cell per leg) is not used.
In one variant of Country Road, not all rooms must be used. Naturally, all cells next to an unused room are part of the loop.
