Question

Five square pyramids are stacked as shown in the figure below with the bottom layer consisting of four pyramids in a 2 x 2 grid. If the bottom layer consists of pyramids in 6 x 7 grid, and they are stacked in a similar manner to its maximum height, how many pyramids will the formation contain?

Hint:

For problems involving series or layered structures, systematically list the values for each layer or term. This reduces the chance of making a calculation error. Double-check the stopping condition for the series (in this case, when one dimension becomes zero).

Answer 1

Step 1: Understanding the Concept:
The problem describes a method of stacking square pyramids in layers. A layer of \(m \times n\) pyramids supports the next layer, which will have dimensions of \((m-1) \times (n-1)\). We need to find the total number of pyramids in a stack that starts with a \(6 \times 7\) base and is built to its maximum possible height.

Step 2: Key Formula or Approach:
The total number of pyramids is the sum of the pyramids in each layer. We need to calculate the number of pyramids in each layer and then sum them up.
Number of pyramids in a layer = \((\text{rows}) \times (\text{columns})\).

Step 3: Detailed Explanation:
Let's list the number of pyramids in each layer, starting from the bottom:


Layer 1 (Bottom): \(6 \times 7 = 42\) pyramids.
Layer 2: \((6-1) \times (7-1) = 5 \times 6 = 30\) pyramids.
Layer 3: \((5-1) \times (6-1) = 4 \times 5 = 20\) pyramids.
Layer 4: \((4-1) \times (5-1) = 3 \times 4 = 12\) pyramids.
Layer 5: \((3-1) \times (4-1) = 2 \times 3 = 6\) pyramids.
Layer 6: \((2-1) \times (3-1) = 1 \times 2 = 2\) pyramids.
\ The next layer would be \((1-1) \times (2-1) = 0 \times 1 = 0\), so the stack stops at Layer 6. The maximum height is 6 layers.
Now, we sum the number of pyramids from all layers:
\[ \text{Total Pyramids} = 42 + 30 + 20 + 12 + 6 + 2 \]
\[ \text{Total Pyramids} = (42 + 30) + 20 + 12 + 6 + 2 \]
\[ \text{Total Pyramids} = 72 + 20 + 12 + 6 + 2 \]
\[ \text{Total Pyramids} = 92 + 12 + 6 + 2 \]
\[ \text{Total Pyramids} = 104 + 6 + 2 \]
\[ \text{Total Pyramids} = 110 + 2 \]
\[ \text{Total Pyramids} = 112 \]

Step 4: Final Answer:
The formation will contain a total of 112 pyramids.