Given a balance scale that is sure to break after X weighings, find an equation for the largest number of coins N, from which you can determine a fake coin that has the wrong weight if
A: You know whether the fake is lighter or heavier
B: You do not know whether the fake is lighter or heavier
(Assume only one of the N coins is fake)
If it is known whether the fake is heavier or lighter, then in X weighings a fake coin can be found among N=3^X coins.
At each weighing, the coins are divided into 3 equal groups. Knowing whether the fake is heavier or lighter ahead of time is enough to determine which group the fake is in.
If the weight of the fake is not known, I will just guess N=(3^X-3)/2 based of the 1(N=0), 2(N=3), and 3(N=12) weighing cases.