Your points relating to the hash function are correct, but the output length of SHA1 is not related, since the secret seed itself is not necessarily the result of a SHA1 hash. It could be longer or shorter than 160 bits.
True, however since we know nothing about the size of the secret seed, we are left with assuming that it could potentially be larger than 160 bits.
I rushed into answering this, and I'll admit that I may have overlooked an important point, but since the secret seed (concatenated with the userID) is passed through the digest function SHA1, it means that there are an infinite number of secret seeds that will all result in the exact same SHA1 value for all the current userIDs. It is only necessary to find 1 of these infinite number of secret seeds to reliably predict when someone will achieve legendary status. As such, the length of the output of SHA1 would seem to be relevant?
In addition, the bottleneck is not so much the number of people that must attain legendary statuseach person currently in the range eligible for legendary status will eliminate roughly half of remaining seeds. So a little more than the bit length of the secret seed of such people should be roughly enough.
I hadn't thought of that. I suppose you're right (I'll have to think about it a bit). Since we don't know the bit length of the secret seed, that could be 8 people, or 10,000 people (or anywhere inbetween). However, as you point out:
The problem is that such elimination requires a huge amount of computation powerbrute-forcing the entire seed space.