 Bitcoin mining and Bitcoin difficulty, so the sort of first explanation that I give to people, the explanation that I give to my mom, the highest level of abstraction, is simply to say that it is necessary for some computer, somewhere, to do a lot of math to make sure that transactions are honest, to make sure that nobody sends the same bitcoins twice. And that usually suffices, and most people understand, okay, I can see that some computers have to do math. And then I say that to incentivize, the network needs to incentivize people to do this, so it was set up so that new bitcoins arrive on the network approximately every ten minutes, and the computers which are successfully doing this math are rewarded for their efforts, are rewarded for their electricity, and that's called Bitcoin mining. You set up your computer to run and verify transactions, and you may get awarded with these new bitcoins that are arriving approximately every ten minutes. If you get a little bit deeper than that, you can see the real genius of Bitcoin, and this is just amazing to me. It's like finding out that anybody can edit Wikipedia. So what type of math are the miners doing? They are solving what are called proof-of-work problems. So a miner sits on the network and gathers transactions, and then wants to add these transactions to the official ledger, which is called the blockchain. And to do that, to add this block of transactions to the blockchain, a miner just needs to find a number that has a very special relationship with that block of transactions. The block of transactions, like anything in computers, can be expressed as a gigantic number, and it finds the number of the block plus this other number when you feed it through a hash function called SHA-256, it'll come out with a really small number. It's called a really special number. And that's it, and that's amazing. The miner is just saying, look, here's the proof that I spent a lot of time thinking about this block. Let's add it to the blockchain, and that's it. And the difficulty speaks to how small the result of the SHA-256 hash function is. In other words, the block I want to add plus my special number fed through the SHA-256 algorithm comes out with a very small number. So the difficulty, if when the difficulty is in, they increase the difficulty by saying the number needs to be this small. Back in the early days, the number only needed to be that small. And that's how the difficulty is adjusted. The difficulty number, I'm not going to go into how it's calculated, but I guess it's just important to note that it has a linear relationship. It's a linear, so a difficulty of 40 is twice as hard as a difficulty of 20. I suppose I should add one thing. A miner by itself cannot add to the blockchain. I mean, it's required for all other miners to accept this block. And if the miners see the block and agree with all those transactions, meaning that they see them as well or don't see anything that contradicts them, then all the other miners will say, okay, the block I was working on, I didn't manage to find a special number first, so we'll put this block and here's the next bunch of transactions. I'll start working on that one. That's if the miner accepts the block proposed by the winning miner, the miner that first found that special number. If the miners do not accept those transactions, if they say, hold on, these don't look like honest transactions, they're just going to keep working. So a dishonest miner, somebody who reprogrammed the miner to insert fraudulent transactions, they have to get lucky to first find a special number first, and then they have to beat the competition again to put another block on top of their block, and all the miners will revert to the longest chain. But if it's just one dishonest block, then all the miners will keep working on theirs without throwing theirs away. And I should note that economics is on the side of honesty, because if somebody has that much hashing power that they can win the race a couple of times in a row and insert double transactions, then they will be more profitable simply using their hashing power to mine bitcoins honestly and make money that way. And even, okay, so let's say someone else gets control of all this hashing power who just wants to ruin bitcoin. Like even then, it's not that big a deal. Like even if they start, I mean it is a big deal, but there's recourse. It's not like the blockchain is obliterated. People still see what the ledger was before those corrupt transactions started entering it. So they could just fork from an old block in the blockchain, say throw away these blocks and we'll revert to this old ownership, or they could patch the code or they could start excluding certain miners. There's all kinds of solutions. So it's not a showstopper. Oh, and the other thing I just wanted to give a shout out to one of the more interesting altcoins called PPCoin. Instead of proof of work, they use something called proof of stake, which I'm not ready to give a detailed explanation about, but the blocks are generated from coin ownership and mining is not such a big deal. You don't need to run all those computers. Really fascinating, really innovative, worth looking into. And that's all. Have a great day.