Estimate the probabilities of finding two messages with the same MD5 checksum, given total numbers of messages of 2⁶³, 2⁶⁴, and 2⁶⁵. This is the birthday problem and again the probability that the (k + 1)th message has a different checksum from each of the preceding k is 1 − k/2¹²⁸. However, the approximation in the hint there for simplifying the product fails rather badly now. So, instead, take the log of each side and use the approximation log(1 − k/2¹²⁸) ≈ −k/2¹²⁸.