Variations of the four cube problem
The four cube problem has been around for over a century under various names such as Katzenjammer, Great Tantalizer, and Instant Insanity. These puzzles have graph theoretic solutions that show that many of them have the same unique solution but with different colors. Unique solutions for the five and six cube problems are known as well. I created a program that can create and evaluate these types of puzzles. My research has shown that there exist cubes that make a solution impossible when they replace any cube in the five and six cube puzzles. There are many different non-isomorphic four, five, and six cube puzzles each with unique solutions that can be verified graphically. Some but not all of them have a no-solution cube as well. This list of puzzles is not exhaustive but shows that there is more than one way to have a unique solution for the five and six cube problems.