Fill the following pyramids: (i) four rows — top 50, second row right box 22, bottom row 4, ?, 6, ? (ii) four rows — second row left box 40, third row right box 9, bottom row 5, ?, 7, ? (iii) four rows — top 35, third row right box 7, bottom row 3, 5, ?, ?
Answer
In each one, call the two missing bottom entries x and y, write down what the given boxes say, and solve.
(i) Bottom row 4, x, 6, y. The right box of the second row sits above 6 and is built from x, 6, y:
(x + 6) + (6 + y) = 22 → x + y + 12 = 22 → x + y = 10
Left box of the second row = 50 – 22 = 28
(4 + x) + (x + 6) = 28 → 2x + 10 = 28
2x = 18 (subtracting 10 from both sides)
x = 9(dividing both sides by 2), so y = 10 – 9 = 1
Bottom 4, 9, 6, 1 → third row 13, 15, 7 → second row 28, 22 → top 50.
(ii) Bottom row 5, x, 7, y. The third-row box marked 9 sits on 7 and y:
7 + y = 9 → y = 2(subtracting 7 from both sides)
40 = (5 + x) + (x + 7) = 2x + 12
2x = 28 → x = 14
Bottom 5, 14, 7, 2 → third row 19, 21, 9 → second row 40, 30 → top 70.
(iii) Bottom row 3, 5, x, y. The third-row box marked 7 sits on x and y, and the top of a four-row pyramid is a + 3b + 3c + d:
x + y = 7
3 + 3(5) + 3x + y = 35 → 18 + 3x + y = 35 → 3x + y = 17
Subtract x + y = 7 from 3x + y = 17: 2x = 10 → x = 5
y = 7 – 5 = 2
Bottom 3, 5, 5, 2 → third row 8, 10, 7 → second row 18, 17 → top 35.
Why it happens: subtracting one equation from another is legitimate for the same reason as before — x + y and 7 are the same number, so taking away x + y from the left of 3x + y = 17 and taking away 7 from the right removes equal amounts from equal sides. What makes it useful is that the y disappears, leaving a single-letter equation you can finish.
What is the relationship between the numbers in the bottom row and the number at the top?
Answer
The top is always a fixed combination of the bottom row — and the multipliers follow a pattern.
Rows
Bottom row
Top
Multipliers
2
a, b
a + b
1, 1
3
a, b, c
a + 2b + c
1, 2, 1
4
a, b, c, d
a + 3b + 3c + d
1, 3, 3, 1
5
a, b, c, d, e
a + 4b + 6c + 4d + e
1, 4, 6, 4, 1
Why it happens: think of a number in the bottom row climbing to the top. At each step it can go up-left or up-right, and it is added into every box it reaches. So a bottom entry is counted once for every path from its box to the top. The two end entries have only one path each, so their multiplier is always 1; the inner entries have many. And the list of multipliers for the next size is made by adding neighbours in the list above — 1, 3, 3, 1 gives 1, 4, 6, 4, 1 — because a pyramid one row taller is just two copies of the shorter one added together.
Did you know? the triangle of multipliers 1 / 1 1 / 1 2 1 / 1 3 3 1 / 1 4 6 4 1 is the famous Pascal’s triangle, known in India centuries earlier as the meru-prastāra of Piṅgala.