Alice and bob can then either do one of 3 (rational!) things - stick to a prior choice, guess their flip result, or guess the opposite of their flip result.
Both of them sticking to prior choice was shown not to work, leaving 8 permutations to investigate at most. But it is easier than that.
I started by assuming Alice will guess her flip result. Looking at the truth table
A B
h h :-)
h t :-(
t h :-(
t t :-)
This means the only losing possibilities are when the coins are different. Therefore Bob must guess the opposite of his flip result, to ensure no losing possibilities at all.
Both of them sticking to prior choice was shown not to work, leaving 8 permutations to investigate at most. But it is easier than that.
I started by assuming Alice will guess her flip result. Looking at the truth table
This means the only losing possibilities are when the coins are different. Therefore Bob must guess the opposite of his flip result, to ensure no losing possibilities at all.