Here are the plans for the color patterns for the three different cubes. Again, each cube had to be unique so that there would only be one possible solution to the total puzzle.
Cutting the smaller cubes themselves was easy - they are just 1.5" cubes, cut on the table saw. The labor-intensive part would be painting all the sides. Since there would be three sides (one for each large cube) in each color, I made a jig to hold twelve smaller cubes tightly together so I could spray paint them. It worked well for the most part. For each new color, I covered the jig itself with painter's masking tape. Sadly, there was some bleed over on some of the edges of the cube, so I did end up having to touch up the edges of most of the cubes by hand with a small brush. Also, I painted the three interior sides of each cube (the sides that won't be visible when the large cube is assembled) black. Here is the cube painting jig:
So, here are all 24 of the colored cubes individually separated, and also put together into the three large cubes:
And, finally, I had to make some sort of holder to keep all the cubes:
This is a simple puzzle but it is more challenging than it looks - especially if you're trying to do it quickly! My record is four minutes.