Who are you solving algorithms for and why? I am sick to death of people responding to my question with logistics of "how to" mine data or just that you are solving "mathematical concepts" and "algorithms."  What I do understand is this - some "organization" seems to think it is worth A LOT of money to have the public data mine for them (i.e.: bit coin payments for solving data). My question is what is this data used for, who is using it, and why?  No one seems to know and anytime I've encountered a secret this well hidden, it usually is nefarious under the surface.
The algorithm of choice is two rounds of SHA256, which has completely arbitrary outputs and a massively unsearchable output space. To this end, there is no data mining or valuable calculation actually being done. Here's what's really happening:

Every so often, the rules of the Bitcoin protocol set a difficulty number. This number sets a sort of target for a value miners are looking for. Since the output of SHA256 is essentially random with respect to its input, miners are all just looking for some random value in a given field that makes SHA256(SHA256(x)) return a value within a specified range. Such a value is considered a "solution" to the mining problem, but really it's less like math and more like throwing darts while blindfolded until you hit the bullseye. The changing difficulty number ensures that block solutions will be found at regular intervals no matter how many people are mining.

This solution does not hold any valuable data or information of any kind, except that it proves that someone somewhere had to have done a lot of work to find a value that produced the desired output. This also means that for someone else to fake that data they would also have to do a lot of work.

Since each block of data references the block before it, these hashes - and the work it took to create them - also stack up, meaning that older data would require a TREMENDOUS amount of computational power to falsify, since you'd also have to falsify every block that came after it, faster than the entire network combined is solving new blocks. This is the entire underlying security mechanism of Bitcoin: Data in the Blockchain represents a consensus and this is the way a network of people with no other trust mechanism reach a consensus.

This is likely why you've had such a hard time finding an answer to your question: There isn't anyone you're solving algorithms for and the only "why" is "because it's hard work, and fraudsters would have to do more hard work than everyone else combined to lie to the network."

Finally, as to the "someone thinks this is worth a lot of money" - there are two sources of mining income. The first is transaction fees paid by those whose transactions are in the mined blocks, which seems pretty reasonable given that the miners' work is the only reason that data will be trustable. The second is the block reward.

One of the more difficult problems to solve when decentralizing a currency is deciding who gets the initial run of the stuff. In this case, it was decided that the miners who are doing valuable work to secure the network should receive that initial issuance, in the form of a "block reward." Every so often this reward is cut in half and eventually it rounds down to zero. At that point all 21 million bitcoins that will ever exist will have been mined and the only reward miners will continue to receive is the transaction fee.