Calculating minimum number of messages hashed a 50% probability of a collision (Birthday Paradox)
You are approximately correct; they are wrong. Their answer calculates the chance of matching a particular value i.e. hash inversion. To see this $k$ tries have a $(2047/2048)^k$ of failing to find a match so you want $k&gt;\log(0.5)/\log(2047/2048)$.
For small numbers we can calculate the exact value of the birthday bound by finding the smallest $k$ such that
$$\prod_{0\le i\le k}\frac{n-i}n=\frac {n!}{(n-k)!n^k}&lt;0.5$$
which in this case is $k=54$.