I'm familiar with the mathematic principles and computer technology behind bitcoin. However, even with reading alot of the beginner's topics and other forums and such, I'm still left with a couple questions about the implementation of the currency.
A few questions:
1. Work done: I understand that my computer (mining at 250 Mhash/s) is chugging away to find a nonce for the hashing algorithm that yeilds a solution that fits in the current difficulty level. However my question is: As I watch the block count go up on the bitcoin client, does my computer have to start from scratch each time?? Or am I still working to find one of the predicted 2016 (or whatever it is) blocks for the current difficulty level?
All miners are working on each block, one at a time. There is a designed delay of about 10 minutes between each block to allow for future network latency of spreading blocks and transactions. There is no "starting from scratch" however, your computer is repeatedly hashing the 80 block header, all miners have equal odds of success with each hash performed.
2. "Accepted": What is a Difficulty 1 hash? And when my computer "finds" one, is it a solution to a block that wasn't under the difficulty target, Or is it some other incremental calculation?
"Difficulty" is a relative metic. It's the number of times more hashes (on average) it takes to find a solution than it would be if the "target" number were it's highest possible setting. The system will not permit a target of greater than 32 leading zeros of the 256 bit hash. Thus a difficulty 1 hash would not solve teh currency difficulty.
3. Remembering: Does my solo miner remember the work it has done if I close it or my computer turns off but I get it back to work within a reasonable timeframe and before the difficulty level goes up?? I ask because the "accepted" number resets each time..
No, and it doesn't need to. It doesn't matter because there is no begining or end to the work.