Here is an alternative solution to the problem of a smaller circle (of radius 1 cm) rotating along the circumference of a larger one (of radius 4 cm). Let us stretch out both the circles into straight lines, placed side by side, with their left ends coinciding. It is easy to see that the shorter line is contained four times within the longer line. This is exactly equal to four rotations of the smaller circle around the larger circle. But, the shorter line is still within the length of the longer line (circumference of the larger circle) since its left end has not coincided with the right end of the longer line. The fifth rotation of the smaller circle is equivalent to the left end of the shorter line coinciding with the right end of the longer line.

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