The grid out-neighbor problem

Welcome to an interactive version of the grid out-neighborhood problem! Here, you can test different configurations of the set \(S\) to see how \(N_{+}(S)\) changes.

The grid square with the red box represents the origin \((0,0)\), which is a type-1 vertex (out-neighbors are left and right, and in-neighbors are up and down). To add/remove a grid square to/from \(S\), simply left-click. The set \(S\) is represented by blue squares, while \(N_{+}(S)\) is represented by greeen squares.