*that last paragraph is from grok explaining your mention of hash trapdoors

oh, the random number? not possible to pick one favorably without the process of mining. small hashes, like used in bitcoin difficulty targets, are very rare. the hash functions used like sha256 or blake2/3 are called cryptographic hashes because they have very very low correlation between inputs and outputs, so-called "collision resistance". by contrast, the hash functions used in machine learning and LLM systems are the opposite, they are very collision prone, and these are generally called proximity hash functions because they allow you to evaluate a distance between related items. the concept of distance between codes was invented/discovered by Hamming, and is the basis of error corrrection systems.

Richard Hamming - Wikipedia

hash functions, collisions, preimages, hamming distances are all related concepts to how bitcoin's security works. they are involved in how asymmetric cryptography (signatures) function as well as hash functions, and are used in PoW, with a difficulty adjustment, to prevent forks in the record and keep it linear. so yeah, you can't shortcut finding a block that satisfies a given target. maybe quantum computers could lower the time required to find these solutions but likely their energy cost to operate would exceed simply playing the game fairly like it was designed to be.

ok, so it's not like the game "I'm thinking of a random number between 1 and 10" and the other person has to guess. in this case, the miner just needs to find some small number that produces a hash with a lot of leading "0's" then i wondered what happens when 2 miners find the small number at the exact same time. research says that the chain forks temporarily, and the nodes resolve to the longer chain. i assume this is in the bitcoin core code that the nodes run.