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I asked a question about mining difficulty in this SO post, Nate explained that the estimated total amout of hashes being calculated to date is 100000000000000000000000000 which is about 0.0000000000000000000000000000000000000000000000001% of the total number of possiblilities.

This brings up my question: If all of the computing power to date has only calculated a tiny small amout of all the possbilities, how can miners still earn their reward every 10 mintues? Does that mean the number of correct hashes (i.e. hashes less than the target) is also large so that bitcoin rewards are given out frequenlty?

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Yes, there are a lot of hashes with low enough values to be acceptable. Precisely, there are target + 1 hashes which are acceptable.

The target varies over time and is adjusted every 2016 blocks (called a "difficulty period") so that with the then-current hashing power of the network, the average block time of the next difficulty period would be 10 minutes.

About every block explorer should be able to show you the block hashes. The hash of the current block (block 453794) is

0x0000000000000000006ec06a9ae920278c462a565cd8f31ea87acaa05e5fbafa

You can see this in the table of the block explorer of your choice:

The difficulty cannot be shown by all block explorers, but by some. It currently is 440779902286.5892. The difficulty merely is an easier-to-read version of the target. You can learn more about it on https://en.bitcoin.it/wiki/Difficulty.

I haven't calculated the current target but it got to be at most

0x0000000000000000006ec06a9ae920278c462a565cd8f31ea87acaa05e5fbafa

because otherwise, the current top block wouldn't be a valid block.

Every hash smaller than that is also accepted. Note that the representation of the hash in in hexadecimal which is the most common way of representing hashes.

You can just ask WolframAlpha what that number is in decimal. It's this monster:

10607898375488084905905799981068947599790252823802788602

Because 0 is also valid, there are at least

10607898375488084905905799981068947599790252823802788603

acceptable hashes.

Or in a more sane notation:

1.06*10^55

For comparison: You're probably made up of somewhere around 7*10^27 atoms and Earth's circumference is about 8.01*10^11 times the width of a human hair. So, yeah ... quite a few options to choose from.

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Mining difficulty adjusts through time. Mining was easier (difficulty was lower) years ago than it is today as you can see on this chart https://blockchain.info/charts/hash-rate

Difficulty adjusts according to the amount of hash power in the network so it takes on average 10 minutes to mine a block. If blocks have been arriving quicker in the last two weeks (2016 blocks), difficulty goes up. If they've been arriving slower, it goes down.

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