I can answer that.

You COULD create a full list of all the solutions... in fact... that's what you are doing by mining.

Miner's computers are performing millions of rounds of a cryptographic hash called SHA256. The thing is, is that the computation you are doing to find these solutions are in fact what gives bitcoin its value. There is literally no other way to do it other than these computational workloads.

The difficulty is what decides "which" solutions are valid. This technically involves finding a solution to the hashing algorithm that falls under a certain number (determined by all the blocks before it). The "number" decreases over time limiting the space of possible solutions thereby making it more difficult.

By design, the system slowly increases the difficulty thereby exponentially making it more difficult to create bitcoins until the system reaches the limit of the binary granularity built it which gives you the arbitrary maximum amount of bitcoins in circulation.