Source: 2018 Canadian Open Math Challenge Part C Problem 1 At Math-$e^e$-Mart, cans of cat food are arranged in an pentagonal pyramid of 15 layers high, with 1 can in the top layer, 5 cans in the second layer, 12 cans in the third layer, 22 cans in the fourth layer etc, so that the $k^{\text{th}}$ layer is a pentagon with $k$ cans on each side. $\text{(a)}$ How many cans are on the bottom, $15^{\text{th}}$, (A.)layer of this pyramid? $\text{(b)}$ The pentagonal pyramid is rearranged into a prism consisting of 15 identical layers. (B.)How many cans are on the bottom layer of the prism? $\text{(c)}$ A triangular prism consist of indentical layers, each of which has a shape of a triangle. (C.)(the number of cans in a triangular layer is one of the triangular numbers: 1,3,6,10,...) (C.)For example, a prism could be composed of the following layers: Prove that a pentagonal pyramid of cans with any number of layers $l\ge 2$ can be rearranged (without a deficit or leftover) into a triangluar prism of cans with the same number of layers $l$.