Alice and her mathematical adventures in Wonderland (Part 1)

Alice tumbled down the rabbit hole and landed on the hard floor of a long corridor. Inside the long corridor were 16 doors, each half the size of the first. Beside the first door was a glass table, on it was a key and a cake. The key can only fit into the 13th door and the cake shrinks Alice at a rate of, one third her current height/crumb. What is the minimum number of crumbs that she needs to consume to fit through the 13th door Assume that Alice is 1 meter tall (she's quite short) and the first door is 4 meters tall.

This is my first problem so please forgive me if it is no interesting