What about a pyramid with three rows?

Using letter numbers for the bottom row, we can write an expression for the top row.

Question

What about a pyramid with three rows? Using letter numbers for the bottom row, we can write an expression for the top row.

Homi · Accepted Answer

Understand the Question

In an addition pyramid, the value of each block is the sum of the two blocks directly supporting it underneath.
For a 3-row pyramid:
- Bottom row: 3 blocks
- Middle row: 2 blocks
- Top row: 1 block
By assigning variables $a, b, c$ to the bottom row, we build up each row algebraically until we obtain the simplified expression for the top block.

Step 1 · Label the Bottom Row

Let the three blocks in the bottom row be labeled from left to right as $a$ , $b$ , and $c$ . Diagram 1

Step 2 · Calculate the Second Row

Each block in the second row is the sum of the two blocks directly below it. Diagram 2

$\text{First block (second row)} = a + b$

$\text{Second block (second row)} = b + c$

Step 3 · Find and Simplify the Top Block

The top block is the sum of the two blocks in the second row: $\text{Top block} = (a+b) + (b+c)$

Combining like terms:

\begin{aligned} \text{Top block} &= a + b + b + c \\ &= a + (b+b) + c \\ &= a + 2b + c \end{aligned}

Answer

$a + 2b + c$

Common Mistakes

Under-counting the middle variable: Forgetting that the middle block $b$ is shared by both blocks in the second row, which contributes $2b$ (not just $b$ ) to the top.
Confusing addition with multiplication: Writing $b + b$ as $b^2$ instead of $2b$ .

Question 10