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https://en.bitcoin.it/wiki/Pooled_mining: A share is awarded by the mining pool to the clients who present a valid proof of work of the same type as the proof of work that is used for creating blocks, but of lesser difficulty, so that it requires less time on average to generate.

I would like to know how the proof of work with lesser difficulty will help in calculating the proof of work for a new block.

In other words, why would pools pay for shares? How do they help?

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In order to find a Block, one needs to perform A LOT of tries - about 2^32 * the current difficulty. That number is so huge that solo miners rarely would be able to solve a block. A share on the other hand is a block of difficulty 1, which is at the moment a few million times easier to find.

Finding a share does not help in finding a block in any way - each hash is independent of all others. However, a share IS a proof of work done by the miner - they put in the effort to help the pool find the block. Once one miner finds a solution to the block problem, the pool earns money and can pay its miners - logically paying everyone for each share submitted. Those shares did not solve the block, but they were a proof that a miner was trying to solve a block. The more shares, the more work needed to be put.

Without shares, everyone would be essentially solo mining - there would be no way to prove that one put in the work, so there would be no reason to be paying the miners - they might be cheating after all.

A useful analogy would be lottery tickets - it is unlikely you will win, but if you live in a small town and decide to share the potential win with everyone that participates in lottery in proportion to how many tickets they bought, you would essentially have a mining pool. Every ticket, even if it does not win you anything, is worth a slice of the winnings - you had to pay for that ticket, so it's a proof that you put in the effort.

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