International Borders Description
International Borders is a puzzle original to this blog to the best of my knowledge. A number of colored circles will appear in the grid. For each color that isn’t gray appearing in the grid, that signifies that there is a region in the grid corresponding to that color (every color for which there is a region is given in the grid, so there are no “hidden” colors). A gray circle belongs to one of the regions, but it is not given to which one it belongs.
The object is to fill in some of the grid squares without circles so that the grid is partitioned into a number of regions equal to the number of non-gray colors in the grid. Any two circles of the same color (including gray circles that belong to that color’s region) should have a path between them through adjacent, unfilled squares, but no two circles of different colors should have such a path. If a circle has a number inside of it, that tells how many of the four adjacent squares are filled in.
Finally, there can be no “useless” filled squares that do nothing to separate the regions. More precisely, any filled square must be adjacent to at least two different regions. (Another equivalent restriction: Erasing any filled square must result in connecting two different regions together.) The solution is unique.
Below is an example puzzle and its only solution.