After reading many articles, I didn't find an answer to the following.
"Miners use special software to solve math problems"
Okay, but for whom? Are there organizations or people that need those math problems to be solved?
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Everyone who uses Bitcoins needs those math problems solved because it's the solution of those problems that secures the blockchain against double spend attacks. – David Schwartz Feb 7 '16 at 11:44
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related: Is there a way to set up proof-of-work systems so they would be even more useful? – Murch♦ Feb 8 '16 at 9:15
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Its not a math problem. Its just search for partial collision in function called sha256.
Sha256 is a famous type of puzzle.
During the writing of this post, a block 397108 was mined.
The solution looks like this: 00000000000000000497b28e9c38673ef8fd2fed317d6eca8a579350b1cba0a6
Notice the zeroes on the beginning. The more zeroes the harder is to find a solution.
The only reason is to waste power to prevent creation of funds from thin air (prevent print too much money) and to confirm payments.
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1Actually the problem is not finding a collision. The correct terminology is a preimage attack, where you are given a target hash value (000...000) and want to find an input that hashes to it. – Nayuki Feb 8 '16 at 5:38
