The problem here lies not in the maths used to get the answer or in incorrect thinking on the part of the problem solver - it lies in asking a vague question. There are several versions of this puzzle, all of them relying on vagueness to trick the reader. There may be several correct answers to the general question "How many rotations did the smaller circle make?":
1: the smaller circle only rotates around the centre of the larger circle once.
4: the smaller circle rotates around its own centre four times.
5: the smaller circle makes two types of rotation as above, totalling five rotations.
undetermined/zero/infinity: no start or end point for either circle has been defined, so the smaller circle rotates indefinitely, or does not rotate at all, because no rotation point has been defined either.
It can be argued that the standard definition for a rotation of a circle is a rotation around its own centre. A revolution, however, is a circular movement around an external point (external to the circle). So, in the above problem, the smaller circle makes 4 rotations and one revolution. (Of course, definitions of rotation and revolution can change depending on the context).
The question cannot be answered - fairly and mathematically - with a single answer, without first defining the rotation point and clarifying the definition of 'rotation' in this instance.
It does not help matters that, in the above example, 'revolution' is used in the diagram, while 'rotation' is used in the general text.
I detest these kinds of 'trick' questions, with the 'gotcha!' at the end. They make people feel foolish when they have only tried to think about the question and answer it with the information provided. If you fail to get the correct answer without being given all the information or when the question is badly or incorrectly phrased, then it is clearly not your fault.
Questions such as these do not support the use of mathematical skills (especially logic) in problem solving - they are essentially interpretative grammar tests, and the correct answer should be spotting the grammar or logic mistakes in the question.
Evidence for the above assertions lie in the fact that even people as intelligent as Yutaka Nishiyama can be duped by them. It is not a difficult problem, but a poorly written one.
It is exactly like the 'Deep Thought' computer in Hitchiker's Guide to the Galaxy providing the answer to the 'Ultimate Question' as 42. When confronted about providing such a flippant, simple answer, the computer states: "I think the problem, such as it was, was too broadly based. You never actually stated what the question was." All poorly stated questions in mathematics should be treated with similar contempt!