Solving the Color Cubes Puzzle 
The diagrams to the right show the color cubes (may be numbered differently from your cubes). 

The graph below uses lines between the colored dots to indicate the colors on the opposite faces of each cube. To distinguish cubes, a different style of line is used for each cube: a heavy line for cube 1, a thin line for cube 2, a dashed line for cube 3, and a dotted line for cube 4. When the four cubes are lined up, four lines of different styles can be used to specify the colors on the tops and bottoms of the cubes. 

For example, the four lines shaded by yellow include all four styles of lines. These four lines would place cube 1 so that the top and bottom colors are blue and yellow (because the heavy line with the yellow shading connects the blue and yellow dots). It would similarly place cube 2 with red and green on the top and bottom, cube 3 with green and yellow, and cube 4 with blue and red. Notice that since each color occurs on two of the lines, we can place the cubes so that each color occurs once on the top and once on the bottom:

If we can find another set of four lines of different styles (all different from the lines shaded yellow), such that each color occurs on two of the lines, we can use these lines to determine the colors on the front and back.
Can you find the other set of four lines of different styles that are different from the yellowshaded lines?