Number Pyramid

How to play Number Pyramid

A number pyramid is a stack of bricks: one at the top, two below it, then three, and so on. The rule is simple: every brick equals the sum of the two bricks directly underneath it. Some bricks are filled in for you. Tap an empty brick and type its number with the keypad or your keyboard (Enter jumps to the next empty brick).

When every brick is filled, any brick that doesn't add up is marked so you can fix it. Every pyramid here has exactly one answer and can be worked out step by step, without guessing. Three rows use small numbers and suit younger players; five rows are a real workout, with totals over a hundred near the top.

How to solve it

• Look for little triangles. Every brick and the two it rests on form a triangle of three numbers. Whenever you know two of the three, the third is a single sum or difference away.
• Add going up. If the two bricks below are 3 and 5, the brick on top is 8.
• Subtract going down. If a brick is 8 and one of the bricks under it is 3, the other must be 8 − 3 = 5. Many pyramids can only be finished by working downwards.
• Each answer unlocks the next. A new number often completes two or three other triangles at once, so check around every brick you fill.

The hidden pattern: Pascal's triangle

Number pyramids hide a lovely piece of mathematics. In a pyramid with four bricks on the bottom row, call them a, b, c, d. The top brick is always a + 3b + 3c + d. With five bricks on the bottom it is a + 4b + 6c + 4d + e. Those multipliers, 1 3 3 1 and 1 4 6 4 1, are rows of Pascal's triangle: each counts how many different paths lead from a bottom brick up to the top.

That makes a handy check. With a bottom row of 2, 5, 1 and 4, the top must be 2 + 15 + 3 + 4 = 24. It also explains why a small change in the middle of the bottom row moves the top so much more than a change at the edge.

Why it's great practice

Number pyramids are one of the friendliest ways to practise adding and subtracting. Each step is a short calculation, but you also have to decide which calculation to do next, and that planning is early algebra in disguise: a missing brick is just an unknown in a tiny equation. For adults, the five-row pyramids are a quick mental-arithmetic warm-up with two- and three-digit numbers.

Where number pyramids come from

Number pyramids are a long-standing favourite of primary-school maths in Germany, where they are called Zahlenmauern ("number walls"), and they appear in classrooms and workbooks in many other countries as addition pyramids or number walls. They need no special equipment, just bricks drawn on paper, which is a big part of why teachers love them.

Frequently asked questions

What is the rule of a number pyramid?

Every brick is equal to the sum of the two bricks directly below it. The bottom row has no bricks underneath, so those numbers are worked out from the bricks above and beside them.

Is there only one solution?

Yes. Each pyramid is generated so that the given bricks lead to exactly one answer, which you can reach one small step at a time without guessing.

Is it suitable for children?

Yes. Three-row pyramids use single-digit bottom numbers and small totals, which is ideal for children learning to add and subtract. Larger pyramids suit older children and adults.

Why do I sometimes need to subtract?

If a brick and one of the bricks under it are known, the other brick under it is the difference. Working downwards like this is often the only way to finish a pyramid.

Is it free?

Yes, completely free with no sign-up. Your best time is saved in your browser, on your own device.