13.3 Number Pyramids

In a number pyramid, every number is the sum of the two numbers sitting directly below it. Take the bottom row 2,\ 7,\ 5:

2 + 7 = 9, \qquad 7 + 5 = 12, \qquad 9 + 12 = 21,

so the pyramid is

\begin{array}{ccccc} & & 21 & & \\ & 9 & & 12 & \\ 2 & & 7 & & 5 \end{array}

Filling from the top down

Pyramids are simple to grow upward, but the puzzles turn interesting when only the top plus a few scattered entries are known and we must work downward. Suppose we are handed the top 14 together with the leftmost numbers 5 and 2:

\begin{array}{ccccc} & & 14 & & \\ & 5 & & ? & \\ 2 & & ? & & ? \end{array}

Since the two boxes in the middle row add to the top, the right-hand middle box is 14 - 5 = 9. The lower-left pair 2 and the unknown add to 5, so the bottom-middle box is 5 - 2 = 3. Finally the bottom-middle and bottom-right add to 9, so the bottom-right is 9 - 3 = 6. The finished pyramid:

\begin{array}{ccccc} & & 14 & & \\ & 5 & & 9 & \\ 2 & & 3 & & 6 \end{array}

When subtraction stalls: call in a letter

Now and then plain subtraction grinds to a halt. Consider this one, with top 80 and bottom-corners 15 and 11:

\begin{array}{ccccc} & & 80 & & \\ & a & & b & \\ 15 & & c & & 11 \end{array}

Not one box can be filled by a single subtraction. So we label the empties a, b, c and write down exactly what the pyramid rule demands:

a + b = 80, \qquad 15 + c = a, \qquad c + 11 = b.

Slotting the last two into the first clears a and b in a single move:

(15 + c) + (c + 11) = 80 \;\;\Longrightarrow\;\; 26 + 2c = 80 \;\;\Longrightarrow\;\; 2c = 54 \;\;\Longrightarrow\;\; c = 27.

Now a = 15 + 27 = 42 and b = 27 + 11 = 38 (and sure enough 42 + 38 = 80 ✓). The pyramid is

\begin{array}{ccccc} & & 80 & & \\ & 42 & & 38 & \\ 15 & & 27 & & 11 \end{array}

Strategy When no box can be settled by a single addition or subtraction, give the unknowns letter-names, convert the pyramid rule into equations, and solve. Algebra reaches places plain arithmetic cannot.

The base controls the top

How does the bottom row decide the apex? Begin with the tiniest pyramid and let the base be letter-numbers a, b:

\begin{array}{ccc} & a+b & \\ a & & b \end{array}

The top is just a+b. Now a three-row pyramid with base a, b, c:

\begin{array}{ccccc} & & a + 2b + c & & \\ & a+b & & b+c & \\ a & & b & & c \end{array}

The middle base number b gets tallied twice at the top. Push on to a four-row base a, b, c, d and the apex becomes

a + 3b + 3c + d.

Those coefficients 1, 2, 1 and 1, 3, 3, 1 are the rows of Pascal’s triangle, a delightful surprise. Each base number’s “weight” at the apex is exactly the count of downward paths leading from it to the top.

Worked Example

Find the apex of a three-row pyramid with base 5,\ 17,\ 9, without building it.

Apex = a + 2b + c = 5 + 2(17) + 9 = 5 + 34 + 9 = \mathbf{48}.

And a four-row pyramid with base 7,\ 21,\ 18,\ 11.

Apex = a + 3b + 3c + d = 7 + 3(21) + 3(18) + 11 = 7 + 63 + 54 + 11 = \mathbf{135}.

Worked Example

A three-row pyramid has apex 63, left base number 8, and middle base number 14. Find the missing right base number.

With base a, b, c the apex is a + 2b + c = 63. Here a = 8 and b = 14, so 8 + 2(14) + c = 63, that is 8 + 28 + c = 63, giving c = 63 - 36 = \mathbf{27}. The base is 8,\ 14,\ 27 (check: 8 + 28 + 27 = 63 ✓). The apex formula works just as well “in reverse” to recover a missing base number.

Figure it Out: Number Pyramids

Practice

  1. Without building the whole pyramid, find the topmost number for each three-row base:  (i) 5,\ 17,\ 9  (ii) 6,\ 13,\ 4  (iii) 11,\ 16,\ 28.
  2. Write an expression for the top of a four-row pyramid in terms of its base a, b, c, d.
  3. Without building the whole pyramid, find the topmost number for each four-row base:  (i) 7,\ 21,\ 18,\ 11  (ii) 9,\ 16,\ 22,\ 8  (iii) 12,\ 8,\ 6,\ 14.
  4. Recall the Virahāṅka–Fibonacci sequence 1, 2, 3, 5, 8, 13, \dots (each term is the sum of the two before it). Place its first three terms 1, 2, 3 in the base of a three-row pyramid. Which numbers appear, and what is the apex? Are they all Virahāṅka–Fibonacci numbers?
    1. Place the first four Virahāṅka–Fibonacci numbers 1,2,3,5 in the base of a four-row pyramid. What do you notice?  (ii) What would happen with the first 29 numbers in a 29-row pyramid?
  6. If the base of an n-row pyramid holds the first n Virahāṅka–Fibonacci numbers, what can you say about the numbers inside, and about the apex?
  1. Apex = a + 2b + c:  (i) 5 + 34 + 9 = \mathbf{48}  (ii) 6 + 26 + 4 = \mathbf{36}  (iii) 11 + 32 + 28 = \mathbf{71}.
  2. Apex of a four-row pyramid = \mathbf{a + 3b + 3c + d} (coefficients 1,3,3,1 from Pascal’s triangle).
  3. Apex = a + 3b + 3c + d:  (i) 7 + 63 + 54 + 11 = \mathbf{135}  (ii) 9 + 48 + 66 + 8 = \mathbf{131}  (iii) 12 + 24 + 18 + 14 = \mathbf{68}.
  4. Base 1, 2, 3 \Rightarrow middle row 1+2 = 3,\ 2+3 = 5; apex 3 + 5 = \mathbf{8}. The grid holds 1, 2, 3, 3, 5, 8, every entry is a Virahāṅka–Fibonacci number, and the apex 8 is the next number in the sequence.
    1. Base 1, 2, 3, 5 \Rightarrow row 2 is 3, 5, 8, row 3 is 8, 13, apex \mathbf{21}. Once more every entry is a Virahāṅka–Fibonacci number, and the apex is the 8th term, 21.  (ii) The same behaviour persists: all entries are Virahāṅka–Fibonacci numbers and the apex is again a (far later) term of the sequence.
  6. Because each pair of consecutive Virahāṅka–Fibonacci numbers adds to the next one, every number in the pyramid is itself a Virahāṅka–Fibonacci number, and the apex is a Virahāṅka–Fibonacci number further down the sequence. The pyramid simply “slides up” the sequence one row at a time.