I haven't reasoned through your solution fully yet, but is that a guarantee that at least one of them is correct? How?
Sorry if your answer turns out to be obviously correct.
Sorry if your answer turns out to be obviously correct.
1. Ava's card is red, and Bruce's is black.
2. Ava's card is red, and Bruce's is red.
3. Ava's card is black, and Bruce's is red.
4. Ava's card is black, and Bruce's is black.
To win, Ava should pick the same color as the card she sees on Bruce's forehead, and Bruce should pick the color that is different from the card he sees on Ava's forehead. (Which player chooses which strategy doesn't actually matter.)
Here is how the strategy plays out for each of the four scenarios above:
1. Ava chooses black, and Bruce chooses black. Bruce guessed correctly.
2. Ava chooses red, and Bruce chooses black. Ava guessed correctly.
3. Ava chooses red, and Bruce chooses red. Bruce guessed correctly.
4. Ava chooses black, and Bruce chooses red. Ava guessed correctly.
I'm unsettled that a strategy involving using information that must have 0 correlation to each person's own correct answer could yield a collectively correct answer. I have heard problems similar to this before where I have been able to say "oohhhhhh" and find out why the answer does not seem to violate other overarching truths, but I'm not there with this one yet.
This is a like a person spelling out, step by step, how to build a perpetual motion machine, with no clear missteps, and the machine works, and I'm trying to understand how it jives with the second law of thermodynamics.
Then how do they arrive at a correctively correct answer? I'm reconciling this by noticing that there is no possibility of them both being correct. That is, their strategy effectively divides up the answer space such that they are covering all two possible correct answers.
What the strategy can influence is the correlation between Ava being right and Bob being right.