This puzzle is very strange. If it's for measuring the world's brute forcing capacity, 161-256 are just a waste (RIPEMD160 entropy is filled by 160, and by all of P2PKH Bitcoin). The puzzle creator could improve the puzzle's utility without bringing in any extra funds from outside - just spend 161-256 across to the unsolved portion 51-160, and roughly treble the puzzle's content density.
If on the other hand there's a pattern to find... well... that's awfully open-ended... can we have a hint or two?

I am the creator.
You are quite right, 161-256 are silly. I honestly just did not think of this. What is especially embarrassing, is this did not occur to me once, in two years. By way of excuse, I was not really thinking much about the puzzle at all.
I will make up for two years of stupidity. I will spend from 161-256 to the unsolved parts, as you suggest. In addition, I intend to add further funds. My aim is to boost the density by a factor of 10, from 0.001*length(key) to 0.01*length(key). Probably in the next few weeks. At any rate, when I next have an extended period of quiet and calm, to construct the new transaction carefully.
A few words about the puzzle. There is no pattern. It is just consecutive keys from a deterministic wallet (masked with leading 000...0001 to set difficulty). It is simply a crude measuring instrument, of the cracking strength of the community.
Finally, I wish to express appreciation of the efforts of all developers of new cracking tools and technology. The "large bitcoin collider" is especially innovative and interesting!
Please if it is possible for us to release the pubkeys from the other addresses that are missing. We are grateful! United We Are One!
Donate: 1C6wmXbktMA1peHMDbDRShAhJ6Wd8Mypw8
What for. You still do not have time to catch anything faster others with rigs. All stay at 120 bits.
***
Continuation of the blind blindbirthday...
Choose (random.choice 2x) from a set (00-99) with a fixed seed()
Find
seed(3435) 0000
seed(108918) 0001
seed(4327) 0002
etc..
from 0000 to 9999 10000 similarly 10000 different seed 3435, 108918, 4327, 20723, 9991, 10068, 8175, 10495, 23048, 5511, 3085 etc... seed [
2532, 679, 4652, 2530, 620] pz
3056 8377 3120 6420 2855
But we have an unlimited number of numbers for fixed results
seed 2532 pz 3056
seed 3801 pz 3056
seed 4476 pz 3056
seed 6363 pz 3056
etc...
seed 1000000000974 pz 3056
seed 1000000006200 pz 3056
seed 1000000072946 pz 3056
seed 1000000074588 pz 3056
etc...
Are there patterns in the units of steps seed or rand?...
But as in the case of a black hole (2^...)
Here is a wormhole
When increasing the number pz will grow and seed
seed 3056
seeed 30568
seeeeed 305683
seeeeeeed 3056837
seeeeeeeed 30568377 etc
and these seed for numbers 10 or 20 digits long too repeated.