How to determine number of alpha hydrogens in a cyclic compound? How do we determine the number of alpha hydrogens in a compound? Where should we start the naming from?
For example, bicyclo[4.2.0]oct-7-ene and bicyclo[4.2.0]oct-3-ene:

Compare the stabilities of the following alkenes


In both these structures, I need to determine their stabilities based on their hyperconjugating structures. I need to determine the number of alpha hydrogens it has, to determine the number of hyperconjugating structures it makes. Where do I start?
I am aware that stability is directly proportional to the number of hyperconjugating structures. More the number of alpha hydrogens, greater is the number of hyperconjugating structures, and hence greater the stability.
The relative stability of your two compounds is not determined by arguments based on hyperconjugation.  Rather the relative stabilities have to do with ring strain.  Placing a double bond in smaller and smaller rings increases the angle strain in the ring system.  A double bond, being roughly $\ce{sp^2}$ hybridized, would prefer to have bond angles around 120°.  This becomes difficult in 3- and 4-membered rings where the internuclear angle is 60° and 90° respectively.  Therefore molecules containing double bonds in 3- and 4-membered rings will be more strained than molecules with the double bond in a larger ring.

Edit: Response to OP's comment

No, the given solution is not the correct answer to the problem.  It is correct that structure B has more alpha (alpha to the double bond) hydrogens (4) than structure A (only 2 alpha hydrogens).  It is also true that hyperconjugation can be used to explain why alkyl groups attached to a double bond act to stabilize a double bond (see this earlier answer for an explanation), and that the double bond in B is more stabilized by hyperconjugation than the double bond in A.  However, hyperconjugation has a much smaller effect on the stability of these molecules than the ring strain effect.