Half And Half

So, I wasn't sure if I should put up a complete solution to the envelope puzzle from last time or just post a partial solution and have follow-up questions, but I have since decided on the latter.

Anyway, a solution to the puzzle (which was also given in the comments, but was my "intended easy solution") is: Alice has two envelopes and flips a coin, filling both envelopes with coins on heads and neither of them on tails. With this, the expected value of a random envelope is $1/2. After Bob finds a removes a coin he knows for a fact the other envelope has a coin, so the expected value of an envelope is again $1/2.

Next a follow-up problem:

Suppose instead of Alice whispering her strategy, you overheard her say "I selected an integer k from 0 to N uniformly at random and placed coins in k of the envelopes." What is the value of N so that the scenario is possible?

Its funny how our brains can work, I was again convinced that this was just impossible, even after knowing the earlier answer, but there is a way to solve this. It even has a unique answer.