The math problems they’re solving is the arithmetic (multiplication/addition/etc) needed to add a set of numbers to the blockchain. The blockchain is just a list of numbers that everyone agrees is the universal list of numbers. By design, the math required to do this takes a lot of arithmetic operations to add a new number. How it does this is a much more complicated discussion that goes beyond ELI5 but this essentially sums it up.
> Is it used for anything?
It’s used to add blocks (numbers representing transactions) to the block chain. That’s it.
> Why are they doing it?
When a miner (the people doing the arithmetic) adds a block, the miner is effectively given a token as compensation. That token is represented by a transaction in the block they just “mined”.
This sounds ingenious and revolutionary until you realize that this ingenious system can cost as much as $60 for a single transaction (Hit this in 2017). Today it’s down to roughly $1, sometimes up to $2, or so. Keep in mind that this cost is essentially measured in the amount of energy it takes to make the transaction. If you put it in the context of how much gas you can buy at the pump, because that’s essentially the cost of the electricity and mining equipment used by miners, you get an idea of how wasteful it is. A credit card transaction is on the same order of magnitude, except the vast majority of that is profit or paid out in salaries to people. Only a tiny fraction of it goes toward the energy or the computing hardware required to support it.
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