Number pyramids

A number pyramid is a triangle of boxes in which each cell equals the sum of the two cells directly below it. If the bottom row is $1, 9, 4$, then the next row is $1+9 = 10$ and $9+4 = 13$, and the top is $10 + 13 = 23$. Forward (bottom to top) is easy. The fun starts when only some cells are filled , and you have to work backwards using algebra.

Concept

The rule. In every pyramid we shall meet:

$\text{(cell)} = \text{(cell below-left)} + \text{(cell below-right)}.$

Filling upward. Given a complete bottom row, just add adjacent pairs to fill the row above, and repeat until you reach the apex.

Filling backward. If you know the top and only some lower cells, the rule gives equations you can solve.

Three-row pyramid: a general formula. Suppose the bottom row is $a, b, c$ and you climb up:

• Second row: $a + b$ and $b + c$.
• Top: $(a + b) + (b + c) = a + 2b + c$.

So the top of a three-row pyramid is always $a + 2b + c$ , the middle term is doubled because $b$ is used twice. This single formula lets you check any three-row pyramid instantly.

Four-row pyramid: another formula. Bottom row $a, b, c, d$. After one step: $a+b, b+c, c+d$. After two steps: $a+2b+c, b+2c+d$. Top:

$a + 3b + 3c + d.$

The coefficients $1, 3, 3, 1$ are Pascal's row. In general, for an $n$-row pyramid, the top equals the bottom row weighted by the entries in the $(n-1)$st row of Pascal's triangle.

Working backwards. When only the top and one of the lower cells are known, set up a letter for the missing entries and solve. Example: top is $60$, bottom is $12, ?, 8$ and the middle row is unknown.

Let the bottom middle entry be $c$, second-row entries $a$ and $b$:

$a = 12 + c, \quad b = c + 8, \quad a + b = 60.$

Substituting: $(12 + c) + (c + 8) = 60 \Rightarrow 2c + 20 = 60 \Rightarrow c = 20$. Then $a = 32, b = 28$. Complete pyramid: bottom $12, 20, 8$; middle $32, 28$; top $60$.

Virahānka-Fibonacci link. If the bottom row is $1, 1, 2, 3, 5, 8, \dots$ (the Fibonacci sequence), then every cell in the pyramid is also a Fibonacci number, because adding two consecutive Fibonacci numbers gives the next one. The top is just a later Fibonacci number , a beautiful overlap of two patterns.

Worked examples

Example 1. Bottom row $6, 2$. Find the top.

• Only $2$ rows; top $= 6 + 2 = 8$.

Example 2. Bottom row $3, 4, 3$. Find the top.

• Second row: $7, 7$. Top: $14$. (Or use formula $a + 2b + c = 3 + 8 + 3 = 14$.) ✓

Example 3. Bottom row $4, 13, 8$. Find the top.

• Formula: $4 + 2(13) + 8 = 4 + 26 + 8 = 38$.

Example 4. Pyramid: top is $50$, bottom row is $4, ?, ?, 6$ and the second row of three has $?, 22, ?$ in the centre. Find the unknowns.

• Let the bottom row be $4, x, y, 6$. Second row: $4+x, x+y, y+6$. The middle entry of the second row is given as $22$, so $x + y = 22$.
• Third row: $(4+x) + (x+y) = 4 + 2x + y$ and $(x+y) + (y+6) = x + 2y + 6$. Sum is the top $= 50$: $(4 + 2x + y) + (x + 2y + 6) = 50 \Rightarrow 3x + 3y + 10 = 50 \Rightarrow x + y = \dfrac{40}{3}.$
• That conflicts with $x+y = 22$, so there is no consistent solution , meaning either the layout or one number is wrong. (This is a useful skill: spotting an impossible setup.)

Try it yourself

1. Bottom row $5, 4, 5, 0$. Find the top.
2. Bottom row $7, 11, 3$. Find the top using the formula $a + 2b + c$.
3. Top $= 60$, bottom row $12, ?, 8$. Find all missing entries.
4. Bottom row $4, 13, 8$. Find the top.
5. Bottom row $10, 14, 25$. Top?
6. Write the algebraic top of a four-row pyramid with bottom $a, b, c, d$.
7. If the bottom row is $1, 1, 2$, what is the top? Recognise this number?
8. A three-row pyramid has a top of $30$ and bottom $a, b, c$ with $a = 5, c = 7$. Find $b$.

Activity

Fibonacci pyramid. Take a $7$-row pyramid. Write the first $7$ Virahānka-Fibonacci numbers ($1, 1, 2, 3, 5, 8, 13$) in the bottom row. Fill the pyramid upward. List every number that appears in the pyramid , are they all Virahānka-Fibonacci numbers? What is the top? How does it compare with the $7$th Fibonacci number?