As a solo miner, what happens to the block that has been assigned to the miner if after a short time he stops mining it altogether.

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The "work" of mining consists of finding a "solution" to a list of transactions. The miners throw in a 'bonus' transaction paying themselves as a reward for solving the problem. But only if they solve their problem "fast enough" do they actually get paid. (Other people are solving their own problem, and there is only one global winner per block. Once a winner is found, everyone starts over on the next block)

In a pool, the master creates a block to work on (giving a bonus to the master) and gives it to the slaves. All the slave workers try to solve that one problem at the same time (adding their power together). If the solution is found in time, the master gets the bonus, and distributes it to the workers in the pool.

So if there are no workers in a pool, obviously no solution can be found, so the master won't get paid.

On the other hand, a "solo miner" doesn't have "assigned blocks". He creates his own problem (giving payout to himself). If he stops working on that problem, he doesn't get paid. Only people who solve the problem (fast enough) can get paid.

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