what the

It is really hard way to solve, but I respect it.
Because eventually you did it.

I like the algo.

Sorry, can you explain me how this works behind the scenes?
toBinary(9) is 1001. And your code computes this correctly as 1001. However I don't understand where this string[0] '1' is being taken from?

1. Step one. LargestPowerOf part - for (9) it is 16.
2. Step two. Binary string will have either +=1 or +=0. +=1 is only when (n-power>=0) condition passes. n is 9, power is 16.
   9-16 is NOT >=0. Hence codition should fail and binary string should get '0' as its first digit. But it still gets '1' somehow.
   Can you explain this to me please?