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^{t h} layer is a pentagon with k cans on each side.
(a) How many cans are on the bottom, 15^{\text {th }}, layer of this pyramid?
(b) The pentagonal pyramid is rearranged into a prism consisting of 15 identical layers. How many cans are on the bottom layer of the prism?
(c) A triangular prism consist of identical layers, each of which has a shape of a triangle. (The number of cans in a triangular layer is one of the triangular numbers: 1,3,6,10, \ldots) For example, a prism could be composed of the following layers:
Prove that a pentagonal pyramid of cans with any number of layers l \geq 2 can be rearranged (without a deficit or leftover) into a triangular prism of cans with the same number of layers l.
