Prisoner A sees 2 white hats
Prisoner B sees 1 white 1 black
Prisoner C sees 1 white 1 black

or

Prisoner A sees 2 white hats
Prisoner B sees 2 white hats
Prisoner C sees 2 white hats

3 white hats, 2 black
2 white hats on head / leaves 1 white and 2 black in the game
(trying to solve)

In case where someone sees black/white hats on opposite sides (B and C see black white), they're not able to solve the answer, because there could be any color on their head. Also, 2 persons could guess. This option is self excluding for persons B and C
In case where someone sees white/white combination, there's still one white left and 2 black, as possibility. Therefore, also unable to solve it. 3 persons have equal chance in solving it, since they all see the same, and can think the same.
Since solution MUST have an answer, the only way is that persons B and C see white/black combination, where A person is the person with black hat.