What we want is functions g(x) and h(y) such that P(g(x)=y && g(y)=x) is maximized.

h(x)=x, g(y)=y give cases

coins guesses

xy g(y)h(x)

HH HH — correct

HT TH — incorrect

TH HT — incorrect

TT TT — correct

So guessing that your partner’s coinflip matches yours results in a 50% success rate. So does guessing that your partner has the opposite.

It is easy to see that this cannot be improved because even if your partner is perfect, you must guess wrong at least 50% of the time.

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