Dear Student,
It seems that there is more to being aromatic than simply a ring with alternating single and double bonds. After considerable development of the underlying theory, the pattern which has emerged is that aromatic characteristics are only expected when there is a ring of pi electrons in which the number of pi electrons is equal to 4n + 2 (where n is an integer, 0, 1, 2, etc.). (This is known as the Huckel rule after its discoverer.) We can check this against the compounds we have considered so far: Benzene has 6 pi electrons (two for each pi bond) which is the number we get from 4n + 2 if n = 1. Cyclooctatetraene has 8 pi electrons, and there is no integer "n" which will make 4n + 2 = 8. Cyclobutadiene has 4 pi electrons and also doesn't fit 4n + 2. There are many other examples which support Huckel's rule.
It is important to be sure that the ring of alternating single and double bonds is complete. If there is an sp3 hybridized carbon in the ring, the conditions for aromatic character are not present, and we do not worry about checking for 4n + 2. Here's an example: