Question

A grocer stacks apples in a pyramid-like stack whose rectangular base is 7 apples by 8 apples. Each apple above the first level rests in a pocket formed by four apples below. The stack is completed by a single row of apples. How many apples are in the stack?

Answer 1

Answer:  168

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Explanation:

If there are n apples in a straight line, then there are n-1 gaps between the adjacent apples.

Having 8 apples will have 8-1 = 7 gaps between the apples.

Similarly, having 7 apples will have 7-1 = 6 gaps between those apples.

The 8 by 7 rectangle in the bottom level will have a 7 by 6 rectangle on the second level. The third level will have a 6 by 5 rectangle, and so on.

Here's a table to keep track of everything

$\begin{array}{|c|c|c|} \cline{1-3}\text{Level} & \text{Dimensions} & \text{Number of apples}\\ \cline{1-3}\text{1} & \text{8 by 7} & \text{56}\\ \cline{1-3}\text{2} & \text{7 by 6} & \text{42}\\ \cline{1-3}\text{3} & \text{6 by 5} & \text{30}\\ \cline{1-3}\text{4} & \text{5 by 4} & \text{20}\\ \cline{1-3}\text{5} & \text{4 by 3} & \text{12}\\ \cline{1-3}\text{6} & \text{3 by 2} & \text{6}\\ \cline{1-3}\text{7} & \text{2 by 1} & \text{2}\\ \cline{1-3}\end{array}$

The last step is to add up everything in the third column

56 + 42 + 30 + 20 + 12 + 6 + 2 = 168 apples total