We define a chain of cubes as a special part of the 3-dimensional cube grid, and on it, we consider the shortest walks from a base vertex. To a well-defined zig-zag walk on the cube chain, we associate a sequence described by a system of recurrence relations and using a special directed graph we determine its recurrence property. During our process, we enumerate and collect some directed shortest paths in the directed graph. In addition, we present two other examples of our graphical method to transform a system of recurrence equations of several sequences into a single recurrence sequence.