Number Pyramids: Find the Top Number Without the Middle

In a number pyramid where each box is the sum of the two boxes below it, the top number is a weighted total of the bottom row, and the weights are fixed. For three layers the weights are 1, 2, 1. For four layers they are 1, 3, 3, 1. For five layers they are 1, 4, 6, 4, 1. So you can jump straight to the top without filling in the middle.

NRICH, the mathematics project at the University of Cambridge, sets the task this way: the numbers on the lower layers determine the numbers above them, so if you are told the bottom layer, can you work out the top number without working out the middle layer? Its questions also include whether changing the order of the bottom numbers changes the top, and how to find the largest possible top from three given numbers. Its follow-up, More Number Pyramids, starts from a single bottom-left number and asks why some tops are impossible.

1. Three layers: top = a + 2b + c

Call the bottom row a, b, c from left to right. The middle row is a + b and b + c. Add those and the top is a + 2b + c. The middle bottom number is counted twice, because it feeds both boxes above it.

Example: bottom row 3, 5, 2. Middle row 8 and 7. Top 15. Shortcut: 3 + 10 + 2 = 15.

This also answers the order question. Swapping the two outer numbers never changes the top. Swapping the middle with an outer number does, unless they are equal.

2. Four layers: top = a + 3b + 3c + d

Bottom row 1, 2, 3, 4 gives a middle row of 3, 5, 7, then 8, 12, then a top of 20. Shortcut: 1 + 6 + 9 + 4 = 20. The two inner numbers count three times each, the outer ones once.

3. Five layers and beyond

For five layers the weights are 1, 4, 6, 4, 1. Bottom row 1, 2, 3, 4, 5 has a top of 1 + 8 + 18 + 16 + 5 = 48, and building it layer by layer gives the same 48. Each new layer adds one more row of the same pattern, where every weight is the sum of the two weights above it. If you know Pascal's triangle, these are its rows.

4. Puzzle 1: the biggest possible top

Place the digits 2, 5 and 9 on the bottom row of a three-layer pyramid in any order. What is the largest top number you can get?

Answer: 25. Put the biggest number in the middle, since it is counted twice: 2, 9, 5 gives 2 + 18 + 5 = 25. The other placements give 21 (middle 5) and 18 (middle 2). All six orders were tried, and 25 is the only maximum (reversing the outer two gives the same value).

5. Puzzle 2: a unique digit row

A three-layer pyramid has three different digits from 1 to 9 in its bottom row, with the left digit smaller than the right digit. The top number is 33. Find the bottom row.

Answer: 7, 9, 8. The top is a + 2b + c, and the largest this can ever be with different digits is 33, using 9 in the middle and 7 and 8 on the outside. With the left digit smaller, the order is 7, 9, 8. Every possible triple was checked, and no other triple reaches 33.

6. Puzzle 3: working backward with the middle layer

In a four-layer pyramid, the row just above the bottom reads 7, 11, 9. All bottom numbers are positive whole numbers, and the bottom-left number is 3. Find the whole bottom row and the top number.

Answer: 3, 4, 7, 2. Starting from a = 3: b = 7 - 3 = 4, then c = 11 - 4 = 7, then d = 9 - 7 = 2. The top is 3 + 12 + 21 + 2 = 38. Building it up: second row 7, 11, 9, then 18, 20, then 38. Without the bottom-left number the problem has four different answers (the bottom-left could be 1, 2, 3 or 4), so one extra clue is needed.

7. Puzzle 4: increasing numbers, top 66

A four-layer pyramid has four different whole numbers on the bottom, increasing left to right, all at most 10. The top is 66. Find them.

Answer: 5, 8, 9, 10. Check: 5 + 24 + 27 + 10 = 66. A full search of every increasing row from the numbers 1 to 10 found exactly one row with top 66.

8. Puzzle 5: three consecutive numbers

The bottom row of a three-layer pyramid holds three consecutive whole numbers. The top number is 48. Find the bottom row.

Answer: 11, 12, 13. If the row is n - 1, n, n + 1, the top is (n - 1) + 2n + (n + 1) = 4n. So the top is always four times the middle number, and 48 means the middle is 12. This also shows that the top can never be 30 for a consecutive row, because 30 is not a multiple of 4.

9. Key takeaways

• A three-layer pyramid has top = a + 2b + c, a four-layer pyramid has top = a + 3b + 3c + d.
• The weights follow Pascal's triangle, and each row is built from the one above.
• To make the top as large as possible, put the biggest numbers where the weights are largest.
• With consecutive numbers on a three-layer bottom row, the top is four times the middle number.
• A pyramid built from sums can be run backward, but you usually need one extra clue.