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From what I understand, we need mining to validate the block, we need the proof of work to limit the number of miners can add a block, and we need the block interval to give time to the new block to spread across the chain.
And usually it's said mining is like a lottery, and hence the alternative idea:

Instead of proof of work, we choose randomly some of the miners registered on the network, only that miner can validate and add the next block, and it has let's say 2min to do it, if not another miner is chosen randomly. In this way we don't need to waste a lot of energy to solve the hash problem.

What is the problem with this solution?
(I know it has a problem, but I can't see it and I think knowing this will help me to understand better why we need the proof of work)

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    Instead of proof of work, we choose randomly some of the miners registered on the network... Who is "we"? – Nate Eldredge Jun 21 '18 at 23:23
  • Instead of proof of work, we choose randomly some of the miners registered on the network... Who controls the registration process? – Pieter Wuille Jun 22 '18 at 2:41
  • @NateEldredge, good point, but maybe there is a way to solve that, like using a random function with seed extracted from previous block hash, it's random and neutral and everybody can check if the node trying to add the block is allowed to do it by the random value. "We" is the software, if we all are using the same software (and that must be case always) then everybody can validate the random node, and nobody can control that random number. I'm still mising something? – Enrique Jun 22 '18 at 11:17
  • @PieterWuille is not that already known? I don't know the details about it, but I guess the Bitcoin network already knows where are the other nodes right? I mean if not what happens when I send a transaction from my wallet? It must send that to some node, and how does it know that node exist and where it is? aren't they registered somewhere? – Enrique Jun 22 '18 at 11:19
  • @Enrique: there is no complete, shared list of all nodes active on the P2P network at any one instant. Each node just has itself a set of peers it is working with, and this set will change as nodes join, disappear, or mis-behave and get dropped. – dbkeys Jun 22 '18 at 13:46

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