The Rolling Cube Problem

Cubes that are freshly painted with blue and yellow paint roll along a table and leave paint behind on the table.

Which of the paint trails above could have been made by a cube whose faces were painted with blue and yellow paint?

For the solution, click "Read More" below.

Solution:

If the cube rolls without changing directions, then four faces will touch the ground, and so the pattern that the cube leaves behind will repeat after four squares.  In this problem, two of the trails have a repeating pattern of 3 squares and are therefore not possible.

In the diagram below, going from top to bottom, the trails are:

Possible, impossible, possible, impossible.