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Number pyramid

Each brick is the sum of the two beneath it. Find the missing one — at the top, in the middle or at the base — as pyramids grow taller and numbers larger.

  • Calculation
  • Untimed
  • Endless levels

How it plays

A pyramid of bricks, each the sum of the two it stands on. One brick is marked ?; work out what it must be. A brick above is a sum; a brick below is the one above it minus its neighbour.

Levels add a row, make the numbers bigger without limit, and hide more bricks (shown as ·), so the ? needs a chain: fill one brick, then use it to fill the next. Every chain is checked to exist before the puzzle is shown.

The wrong options include adding where you should subtract. The working walks the chain, one brick at a time.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This practises addition and subtraction inside a structure, and planning which brick to fill first. It is practice at these puzzles, nothing wider.

Learn

How to get better at number pyramid: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Look at the three bricks the ? belongs to: the two beneath it, or the one above it and its neighbour. If two of them are known, you are one step away.
  2. Going up is adding: a brick is the sum of the two it stands on. Going down is subtracting: a brick is the one above it minus its neighbour.
  3. If the ? needs a brick marked ·, find a brick you can fill now that sits in one of the ?’s triples. Fill it first, then the ?.
  4. Say which way you are going — up or down — before you do the sum. The commonest wrong answer is the right two numbers combined the wrong way.

Techniques and tricks

  • Up adds, down subtracts

    For a brick above, add the two below. For a brick below, take its neighbour from the brick above.

    e.g. Above 14 and 12 sits 26; beneath 26, beside 14, sits 26 − 14 = 12.

  • Pascal’s weights

    The top of a pyramid is its bottom row weighted by a row of Pascal’s triangle: 1 2 1 for three rows, 1 3 3 1 for four, 1 4 6 4 1 for five, 1 5 10 10 5 1 for six. You can skip the middle rows.

    e.g. Base 3, 5, 2: the top is 3 + 2 × 5 + 2 = 15 (and 8 + 7 = 15).

  • The same weights for any brick

    Any brick is the bricks beneath it, in the triangle below it, weighted the same way. A brick two rows up is a + 2b + c of the three under it.

    e.g. Over 4, 6, 1, two rows up: 4 + 12 + 1 = 17.

  • A gap in the base from the top

    With the top and all but one base brick known, the weights give the missing one: take the others’ weighted sum from the top and divide by its weight.

    e.g. Three rows, top 15, base 3, ?, 2: 15 − 3 − 2 = 10, and 10 ÷ 2 = 5.

  • Last-digit check

    The last digit of a sum or difference depends only on the last digits. Check your answer’s last digit before you choose.

As the levels climb

  • Rows come first: 3 at level 0, 4 at level 1, then 5 and 6 by level 7, up to the “Most rows” setting (6 unless you change it, 8 at most).
  • The base numbers grow by half again each notch, without end: up to 5, then 8, 11, 17, 25, 38 and on. The bricks near the top grow fastest, so expect three- and four-digit sums high up.
  • Hidden bricks (·) arrive at level 3, one more at a time up to four. Then the ? needs a chain; plan it from the ? outwards before you add anything.
  • In tall pyramids, Pascal’s weights let you jump several rows at once instead of filling every brick on the way.

Traps to avoid

  • Adding where you should subtract: the brick above plus its neighbour, given for a brick below. If your answer is bigger than the brick above it, you went the wrong way.
  • Subtracting where you should add: the difference of the two below, given for a brick above. A brick is always bigger than both bricks it stands on.
  • One out or ten out, from a slip in the sum. The last-digit check catches the first; a rough size check catches the second.

How to practise

  • Use “Bricks asked for” to practise one kind: “A bottom brick” is all subtraction, “The top brick” all addition, “A middle brick” a mix.
  • Lower “Most rows” to 4 while you learn the chains, then raise it to 8 to make Pascal’s weights pay.
  • Switch Answer by to “Typing the number” once the options feel easy; the options let you spot the wrong-direction lure without finishing the sum.

Levels

It is untimed unless you turn the clock on. There is no last level: at least one dial turns without end. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

More calculation

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