No Sound

I have a problem with my iPhone 4 that I've been trying to figure out for a few months but have been unsuccessful so far.

All of a sudden, I lost sound on most applications. The only time the iPhone 4 would make a sound was when you called it. It would ring.

I can also go to Settings > Sounds > Ringtone and then select different ringtones. The iPhone would play those ringtones.

But that's it. Any other native or third-party applications will have no sound.

I can plug in a headphone and hear sound. But the iPhone 4's speakers will otherwise not work.

This happened when the iPhone had iOS 5. I tried to reset the phone to factory settings. I also tried to restore it but the problem was not solved.

When I upgraded to iOS 6, the problem persisted. In iOS 6, I tried to reset the phone. I also restore it but had no luck.

I even tried to jailbreak it but still wasn't able to solve the problem.

I mean, it may be a hardware problem but why will the phone ring if the speaker is completely broken? Or why can I make it sound by going to the settings and playing with ringtones?

Any ideas?