1. "Why did Albert speak first?"
Before we answer this, we need to develop a set of assumptions to operate under. Given the synthetic nature of the problem, we might assume that both Albert and Bernard are of equal ability, and able to make logical inferences based on reasonable assumptions. We'll also assume that both Albert and Bernard are only going to announce the binary state of each others certainty of the answer. Lastly, we'll assume that whoever can deduce the binary state of certainty first, will speak first.
These assumptions, while a bit presumptuous, seem on the surface to adhere to the spirit of the puzzle.
Given those assumptions, what does it mean when Albert speaks first? It means that he's figured out the binary state of certainty before Bernard. How could that be?
If Bernard has the dates 18 or 19, then he knows that he knows, and he also knows that Albert does not know. If Bernard has any other date, he'll know that they both don't know. How does Bernard make this determination? He checks if his date is repeated anywhere.
Compare that to the logic that Albert must perform to rule out Bernard's certainty. He must check that all of the dates in his month are in fact repeated.
If they both perform these mental operations at the same speed, then Bernard should speak first in the case where he knows a unique date. The only reason why Bernard might not speak first is that he must also reason through whether or not Albert might know the date at this point. If Bernard holds an unrepeated date, the complexity of reasoning through Alberts situation boils down to considering only one month. But if Bernard holds a repeated date, he must consider Albert's situation for two months.
So, there exists a time past which Albert must know that Bernard is considering the more difficult situation, and thus he can infer that Bernard does not know.
If one assumes precise knowledge of the timing of logical operations, one can make even stronger inferences. To the point of even solving the entire problem without anyone every saying anything, for specific birthdays.
I will be the first to admit that this line of reasoning may require certain assumptions which are strained. Thus the question may still remain:
1. "Why did Albert speak first?"
Does anyone have a different set of logical assumptions which leads to Albert speaking first?
----
What this exposes really, is the more subtle assumption that the "correct" solution makes:
"There is only one way to make inferences in this puzzle."
The truth of that statement depends very much on the assumptions one makes about the puzzle. Most problematic is that most assumptions which makes that true, make it impossible for Albert to be the first speaker.
It's a kind of logical paradox brought on by the fact that synthetic logic problems do not map very well to real world situations.
The speed at which they come to their conclusions, and who speaks first is meaningless. I think perhaps if you are taking them into account then you are trying to come up with a completely different problem, or intentionally being disingenuous.
Given the context (that this is a question on a high school test), it is safe to assume that Albert and Bernard are only drawing logical conclusions from the information Cheryl provides and what each other say.
Given that the problem works with no problems under these assumptions, and that there is enough information to re-word it with Bernard speaking first that does not change the answer, it seems pointless to try and look for timing attacks or other outlandish ways to change the answer...
That's your own assumption. And that's a perfectly fine assumption to make, but not a universally chosen one.
Maybe my other comment was right and they should have been called Alice, Bob, and Carol.
The way I read the question, it's as though a moderator said: "Albert, tell me what's on your mind." "Bernard?" "Albert?"
I will add that #4 should certainly be in the past tense ("at first I didn't know") because this way Albert and Bernard are in on the fact that they're part of a logical puzzle!! They should not know that they are characters in a puzzle.
This is how I read the puzzle:
Moderator to Albert: tell the reader what you know.
Albert: I don't know when Cheryl's birthday is, but I know that Bernard does not know too.
Moderator to Bernard: And you, Bernard?
Bernard: At first in this puzzle I don't know when Cheryl's birthday is (before Albert spoke), but now that you have asked Albert, and I've heard his response, I've figured it out.
Moderator to Albert: And any thoughts from you, now that Bernard has figured it out?
Albert: Yes. Given the information Bernard has now stated, I would say I also know Cheryl's birthday now.
If you're even thinking about this then you have failed the question, it's a puzzle that asks you to derive a correct answer from the data provided, not from your conjectures about human behaviour.
