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Instead of a maths puzzle, why can't we have miners collect transactions, then just make them wait 10 minutes before accepting the first block broadcasted?

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Proof of Work (the proof that "maths puzzle" was solved) serves a variety of purposes. The primary purpose being to determine the "first" miner whose block should rightfully be accepted as the next block in the chain. If there was no cost to proposing the next block, there would be far too many candidate blocks and the network would split too often.

With Proof of Work, the network can determine which miner has earned the right to mine the next block by having expended the electricity to produce the hash that meets the difficulty level.

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Because, although it would be possible to create a system that worked like this, it would result in a massive amount of orphan blocks and chain splits and, it would not be clear to most nodes which is the longest chain. Think of a blockchain structure like The Ganges river system.

The Ganges

Every ten minutes, 10,000 new blocks would be distributed if there are 10,000 miners. Each of these 10,000 blocks would be at the same blockheight so that it is not possible to determine the longest chain. The longest chain is the block that is next built upon but, after ten more minutes all miners distribut another block building on the block that they prefer. There are now 10,0002 possible blockchains (although less than this will actually be created) and, none of them are yet the longest. This process continues.

  • Makes a lot of sense – Kyle Graham Mar 17 '18 at 1:14

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