Is there an algorithm I can use to unlock an HTC, Samsung, etc. phone instead of paying for an emailed unlock code? (like for Nokias) If not, how do these pay-sites generate their codes?

There *is* an algorithm that can do that. There are also algorithms to generate serial numbers for software like Windows or some games. There is also an algorithm that generates credit card numbers. There are less than 10^16 Visa cards, most 16-digit numbers are NOT valid card numbers, yet they are not generated randomly or sequentially; they use an algorithm. But *YOU* can't use it. That's kind of the point. Many hackers spend much time and risk many years in prison to either hack into companies to obtain those algorithms or reverse-engineer them to code a keygen or similar. Some places are licensed to use that algorithm. Ordinary people are not, and using a keygen can get them in trouble.

Is there a way, the answers no. The sites generate solutions, I have heard of companies spending up to $250,000 to create the solution, so they don't release it easily, you have to give them a code.