There is a machine that turns rocks into diamonds using these simple rules

i). If n = 1, 2, 3, 4, 5, 6 then the machine returns n diamonds ii). If n % 3 = 0 then the machine replaces n rocks with n/3 rocks and returns to i) iii). If n % 3 = 1 then the machine takes 5 rocks and returns to i) iv). If n % 3 = 2 then the machine adds 12 rocks to the rocks already inside the machine and returns to i)

Suppose you have as many rocks as you need and in each problem given you are allowed to give the machine rocks only once. Please show working when answering the questions.

a). if this machine gives you 2 diamonds transforming n rocks, then n is not a power1 of 3. Use contrapositive to prove this claim.

b). Disprove the claim It is not possible to make this machine run forever by finding a counterexample.

c). There is some number of rocks such that the machine will give you exactly 4 diamonds after transformation. Prove this claim by construction.

d). Prove the claim The only way to get 1 diamond is to feed the machine 1 rock (Hint: it is A B proof.)