Wow first time using this forum so I didn't expect an answer as thorough as scientia's. I wish I had more bonus points to give out.
Julianthemath that is the correct way to start, there are three patterns to be found depending on the initial three digit value you subtract.
The way I tackled this so far was to say √5 = a/b
so 5 = (a/b) (a/b) or 5x (a²/b² )
5b² = a²
So let a = 5x
a² = 25x
25x = 5b²
5x = b²
I have no idea if this is the right approach but it made sense when I initially wrote it, seems a bit iffy looking back as I dont really know where I was going with this. I think I was going to look for even numbers of prime factors but cant recall why.
Can you notice any patterns in the sums of the digits in three digits numbers and the sums of the digits of their complements to 1000.
For example 1000 - 255 = 745
2+5+5 = 12
7+4+5 = 16
12+16 = 28
Bonus points for multiple patterns,
Bonus bonus points for anyone who can find a direct proof.