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What happens when two miners (PoW era) find the exact same solution to mining an identical block, and broadcast the solutions at the same time? I'm wondering if a miner could forge uncles this way in order to get extra reward almost every time it successfully solves a block. Imagine that a mining pool finds a solution to a block, and it broadcasts it twice as two different identities (miner nodes). Are(were) there any mechanisms to prevent this in the protocol?

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In this case, the blockchain forks and temporarily has two heads. The nodes will accept the first block they've seen based on its block timestamp until another block is mined, making one of the chains longer, and then resolves the chain fork (the nodes working on the shorter chain will throw out these blocks and the transactions within these blocks will be recycled into the mempool so these transactions will have another chance to be confirmed onto the longer chain).

A miner could forge uncles, but not all uncles are rewarded. In Ethereum, one block can include up to two uncles. If a block references two uncles, each uncle will earn 7/8th of 5 ETH to whoever mined that uncle. A stale block (a block that is not part of the main chain) can only be included as an uncle by up to the seventh-generation descendant of one of its direct siblings, and not any block with a more distant relation. Therefore, although miners could forge uncles, block time should not encourage more uncle blocks than what can be rewarded.

See: https://medium.facilelogin.com/the-mystery-behind-block-time-63351e35603a

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