proof of work – Is it always possible to find a number whose hash starts with a certain number of zeros?

The core principle behind the hash function, like SHA-256, is that it generates a seemingly random output for any given input and the output is uniformly distributed across the possible range. This means that for any given input, the chance of a particular bit being 0 or 1 is approximately 50%. Therefore, the chance of the first bit being zero is 1/2, the chance of the first two bits being zero is 1/4, the first three bits being zero is 1/8, and so on.

Due to these characteristics, it is statistically almost certain that a valid input exists which will generate a hash with a specific number of leading zeros. It is just a matter of trying enough inputs (or “nonces” in the case of ₿ mining) until you find one. This is what ₿ miners do in the proof-of-work system – they try billions of different inputs every second until they find one that generates a hash with the required number of leading zeros.

What would happen if there was no such number?

The ₿ network would be unable to produce a new block for the current difficulty level.