How to Solve a Number Pyramid, Step by Step
A number pyramid is solved by one rule applied over and over: each brick equals the sum of the two bricks directly below it. That gives you two moves, add and subtract, and nearly every pyramid can be finished by choosing the right one at each step.
Below is a worked example, a short method for deciding which move to use, and what changes when pyramids get harder.
The one rule
In a pyramid with four rows, the top brick sits on two, those two sit on three, and those three sit on four. Take any brick that has two bricks under it:
brick = left brick below + right brick below
That is all. Everything else in this guide follows from it.
Worked example
Here is a four-row pyramid. Some bricks are given, and the rest are marked with a question mark.
| Row | Bricks |
|---|---|
| Top | 34 |
| Row 2 | 17, ? |
| Row 3 | ?, 7, ? |
| Bottom | 6, ?, 3, 7 |
This is an example in the style of an Easy puzzle, not a page from the book. Here is how to solve it.
Step 1: Look for a brick you can find by adding. The bottom row has 3 and 7 side by side. The brick above them is 3 + 7 = 10. Write 10 in the right end of Row 3.
Step 2: Look for a brick you can find by subtracting. Row 2 has 17 on the left and a question mark on the right. The top brick 34 is the sum of those two, so the right brick is 34 − 17 = 17.
Step 3: Use the new brick. Row 2's right brick, 17, is made of the two Row 3 bricks under it: the middle one (7) and the right one (10). 7 + 10 = 17. It checks out.
Step 4: Work left. Row 2's left brick, 17, is made of the left and middle bricks of Row 3. The middle one is 7, so the left one is 17 − 7 = 10.
Step 5: Finish the bottom. The left brick of Row 3 is 10, made of the first two bricks on the bottom row. The first is 6, so the second is 10 − 6 = 4. Check the next brick over: 4 + 3 = 7. It matches.
The finished pyramid:
| Row | Bricks |
|---|---|
| Top | 34 |
| Row 2 | 17, 17 |
| Row 3 | 10, 7, 10 |
| Bottom | 6, 4, 3, 7 |
The method in one table
When you are stuck, look at each empty brick and ask what its neighbors give you.
| What you can see | What to do |
|---|---|
| Both bricks directly below are known | Add them |
| The brick above is known and one brick below is known | Subtract the known one from the brick above |
| Neither of the above | Skip it and come back |
Work through the pyramid in rounds. Each round fills at least one brick, and the new brick unlocks the next.
Three habits that prevent mistakes
- Check each brick you fill. After writing a number, add the pair that makes it, or the pair it belongs to, and see that the sum is right.
- Fill in light pencil. If a number leads to a contradiction later, you can erase without a mess.
- Keep to whole numbers. In these puzzles all bricks are whole numbers and never negative. If you get a negative or a fraction, one of your earlier bricks is wrong.
What changes in harder pyramids
The rule never changes. What changes is how many bricks you are given and where.
- Easy: four rows, with almost the whole base given. You mostly add your way up.
- Medium: five rows and bigger numbers. You begin to subtract your way down.
- Hard: six rows with clues up high. You work backward from the top and chain several subtractions before you can add anything.
In the Number Pyramid book in PuzzlePath Math Puzzle Pack, the 50 puzzles follow that path. Puzzles 1–15 are Easy (grades 3–4), 16–35 are Medium (grades 5–6) and 36–50 are Hard (grades 7–8). Each puzzle is checked by a separate solver program that confirms it has exactly one answer, so you never need to wonder whether a different filling would also work. The book starts with a "How to play" page, a worked example, and ends with an answer key. It is one of seven games in the pack, and it practices addition, subtraction, number sense and missing addends, which is the groundwork for algebra.
Why pyramids are good practice
A number pyramid hides a missing-addend problem inside a puzzle. When you work out that 17 − 7 = 10, you are solving 7 + ? = 17 without writing an equation. It is also a satisfying puzzle with built-in checks, which matters more for keeping a child going.
Try this next
If you like pyramids, try crack the code math puzzles, where shapes hide numbers, or math mazes. To see which puzzle suits which grade, read math puzzles for 3rd to 6th grade, and for the full range of seven game types, see printable math puzzles.