Advanced fake-coin problem There are n ≥ 3 coins identical in appearance; either all are genuine or exactly one of them is fake. It is unknown whether the fake coin is lighter or heavier than the genuine one. You have a balance scale with which you can compare any two sets of coins. That is, by tipping to the left, to the right, or staying even, the balance scale will tell whether the sets weigh the same or which of the sets is heavier than the other, but not by how much. The problem is to find whether all the coins are genuine and, if not, to find the fake coin and establish whether it is lighter or heavier than the genuine ones.

a. Prove that any algorithm for this problem must make at least [log3(2n + 1)] weighings in the worst case.

b. Draw a decision tree for an algorithm that solves the problem for n = 3 coins in two weighings.

c. Prove that there exists no algorithm that solves the problem for n = 4 coins in two weighings.

d. Draw a decision tree for an algorithm that solves the problem for n = 4 coins in two weighings by using an extra coin known to be genuine.

e. Draw a decision tree for an algorithm that solves the classic version of the problem-that for n = 12 coins in three weighings (with no extra coins being used).