We have an integra that has intermittent starting problems.

When i tried to crank it sometimes it starts and sometimes it doesn't.
It just ticks once.

So, i changed the battery, twice but the problem still exists.

Most of the times now when i crank, it ticks once, like there's no battery. but battery is full. i'll then try to crank again, and it starts and car runs smoothly. sometimes i have to try 3-4 times. other than that, the car runs perfectly.

any ideas what the problem is?
starter motor, crank angle sensor, alternator, distributor car, sparks?

my understanding are the above problems have different symptoms.

thanks