3) Not all nodes try to solve the puzzle, but all are checking new blocks for validity. Node vs miner. That's why it's so important to run our own nodes, even when not mining. Miners work for us, they get paid to secure the network.
But all nodes control the work of the miners.
Can't stress that often enough.
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I could have clarified miners versus nodes a bit better. As I went through this again, I felt Satoshi was purposely a bit vague in differentiating the two, so as to keep the paper concise.
But I agree, miners do the mining and nodes do the verifying. Nodes might be considered more advanced than miners. Miners do a complex task: working on the puzzle. But it is only one single task.
Nodes have a bit more complexity in their role at verifying all the transactions, and keeping a copy of the blockchain, that is constantly updated.