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Cambridge Primary Checkpoint · Mathematics 0096 · Stage 6 · Number

Square Numbers and Cube Numbers

Know the square numbers and the cube numbers from 1 to 125, link cubes to squares, and use square numbers to find terms in a sequence from their position.

British & IGCSE › Checkpoint 6 › Checkpoint 6 Maths — Complete Course

In the curriculum
  • Use multiplication and what you know about square numbers to recognise cube numbers from 1 to 125 (6Ni.08).
  • Use square numbers to work out terms in a sequence when you are told the position, for example "square the position number and add 3" (6Nc.04).

A square number is what you get when you multiply a whole number by itself. A cube number is what you get when you multiply a whole number by itself and then by itself again. The names come from shapes: square numbers make squares of dots or tiles, and cube numbers make cubes of small blocks.

Key words

Word / phraseMeaning / use
square numberعدد مربع — a number made by multiplying a whole number by itself: 7 × 7 = 49
squared (²)تربيع — 7² is read "7 squared" and means 7 × 7
cube numberعدد مكعب — a number made by multiplying a whole number by itself three times: 4 × 4 × 4 = 64
cubed (³)تكعيب — 4³ is read "4 cubed" and means 4 × 4 × 4
sequenceمتتابعة — numbers in order that follow a rule
termحد — each number in a sequence
positionالترتيب — where the term is: 1st, 2nd, 3rd …
position-to-term ruleقاعدة من الترتيب للحد — a rule that turns the position number into the term

1. Square numbers

1 × 1 = 11²2 × 2 = 42²3 × 3 = 93²4 × 4 = 164²5 × 5 = 255²
The first five square numbers drawn as squares of tiles: 1, 4, 9, 16, 25.

Learn the square numbers up to 12 × 12 by heart. They come up again and again (areas, fractions, sequences).

n123456789101112
n²149162536496481100121144
  • Pattern: the gaps between square numbers are the odd numbers: 1, 4, 9, 16, 25 go up by 3, 5, 7, 9 … So the next square after 25 is 25 + 11 = 36.
  • Last digits: a square number can only end in 0, 1, 4, 5, 6 or 9. So 47, 82 or 93 can never be square numbers.
  • Factors: a square number has an odd number of factors, because one factor pairs with itself (36 = 6 × 6). Every other number has an even number of factors.
  • Bigger squares: use multiplication: 20² = 20 × 20 = 400; 15² = 15 × 15 = 225.

2. Cube numbers (1 to 125)

Build a cube from small cubes. A cube with edges 3 cubes long has 3 layers, and each layer is a 3 × 3 square of 9 cubes. So 3³ = 9 × 3 = 27. This is the link with square numbers: n³ = n² × n.

1³ = 12³ = 83³ = 274³ = 645³ = 125
The cube numbers from 1 to 125: 1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125.
CubeAs a multiplicationUsing the squareCube number
1³1 × 1 × 11 × 11
2³2 × 2 × 24 × 28
3³3 × 3 × 39 × 327
4³4 × 4 × 416 × 464
5³5 × 5 × 525 × 5125
  • 1 and 64 are both square and cube numbers in this range: 64 = 8² = 4³, and 1 = 1² = 1³.
  • Cube numbers grow very fast: from 64 to 125 is a jump of 61.
  • Do not mix up the small number with the multiplication sign: 4³ means 4 × 4 × 4 = 64, not 4 × 3 = 12. And 5² = 25, not 10.
بالعربي: المربع والمكعب
  • العدد المربع = العدد × نفسه. مثال: 6² = 6 × 6 = 36 (مش 12).
  • العدد المكعب = العدد × نفسه × نفسه. مثال: 4³ = 4 × 4 × 4 = 64 (مش 12).
  • احفظ المكعبات: 1، 8، 27، 64، 125. وطريقة سهلة: المكعب = المربع × العدد (5³ = 25 × 5 = 125).

3. Square numbers in sequences

A position-to-term rule tells you how to get any term straight from its position number, without writing all the terms before it. Some rules use square numbers.

Position1234510
Position²1491625100
Position² + 347121928103
Position² − 1038152499
2 × position²28183250200

How to recognise a square-number sequence: look at the gaps. If the gaps are 3, 5, 7, 9 … (going up by 2 each time), the sequence is linked to the square numbers. Then compare each term with the square of its position.

  1. Write the position numbers 1, 2, 3, 4, 5 above the terms.
  2. Write the square numbers 1, 4, 9, 16, 25 under them.
  3. Find what you do to each square number to get the term (the same every time).
  4. Write the rule in words: "square the position number, then add 3".
  5. Use the rule for any position: the 10th term is 10² + 3 = 103.
Pattern 1Pattern 2Pattern 3
Pattern n is an n × n square of blue tiles with 1 yellow tile on top. Tiles = position² + 1: 2, 5, 10, …

In this tile pattern, Pattern 3 has 3 × 3 = 9 blue tiles and 1 yellow tile = 10 tiles. The rule is tiles = position² + 1, so Pattern 10 would need 10² + 1 = 101 tiles.

Worked examples

Worked example 1 (1 mark) — Ring all the square numbers: 12 36 48 64 81 90

36 = 6², 64 = 8², 81 = 9². Answer: 36, 64, 81. 12, 48 and 90 are between square numbers (e.g. 49 = 7² is next to 48). Common mistake: ringing 12 because 12 is "in the 12 times table" — square numbers are n × n, not just multiples.

Worked example 2 (1 mark) — Work out 4³.

4 × 4 = 16, then 16 × 4 = 64. Answer: 64. Common mistake: 4 × 3 = 12.

Worked example 3 (1 mark) — Write a cube number that is between 30 and 100.

Cube numbers: 1, 8, 27, 64, 125. Only 64 is between 30 and 100. Answer: 64.

Worked example 4 (2 marks) — The rule for a sequence is "square the position number and add 3". (a) Write the first four terms. (b) Work out the 10th term.

(a) 1² + 3 = 4, 2² + 3 = 7, 3² + 3 = 12, 4² + 3 = 19. Answer: 4, 7, 12, 19. (b) 10² + 3 = 100 + 3 = 103. Common mistake: (2 + 3)² = 25 — you square the position first, then add 3.

Worked example 5 (2 marks) — Here is a sequence: 0, 3, 8, 15, 24, … Find the position-to-term rule and the 8th term.

Position12345
Square of position1491625
Term0381524

Every term is 1 less than the square: term = position² − 1. 8th term: 8² − 1 = 64 − 1 = 63. Check the gaps: 3, 5, 7, 9 go up by 2, so a square-number rule makes sense.

Worked example 6 (1 mark) — Using the tile pattern above (n × n square plus 1 tile), which pattern number uses 50 tiles?

Tiles = position² + 1 = 50, so position² = 49 and position = 7 (7 × 7 = 49). Answer: Pattern 7. This is working backwards (inverse): subtract 1, then find the number that squares to give 49.

Common mistakes
  • Reading 5² as 5 × 2 = 10 or 4³ as 4 × 3 = 12. The small number tells you how many times the number is multiplied by itself.
  • Thinking 9 or 27 is a cube number because it is in the 3 times table. Only 1, 8, 27, 64, 125 are cubes up to 125 — check by multiplying.
  • Squaring after adding: for "position² + 3" square first, then add.
  • Using a term-to-term rule (+3, +5, +7 …) for the 20th term — it is slow and easy to get wrong; use the position-to-term rule.
  • Mixing up the position and the term in a table.
How Checkpoint asks this
  • "Ring the cube numbers" / "Write a square number between 50 and 70" — know 1–144 squares and 1–125 cubes by heart.
  • "Work out 3³ + 4²" — calculate each part first: 27 + 16 = 43.
  • "The rule is … Find the 10th term." — substitute the position number.
  • "Here is a sequence … Find the rule." — compare each term with the square of its position.
  • Tile or dot patterns — count the tiles in each pattern, make a table, then look for squares.
  • "Is Ali correct? Explain." — show the correct multiplication (4³ = 4 × 4 × 4 = 64).
  • Venn or Carroll diagrams with "square number" / "cube number" labels — remember 1 and 64 are both.

Practice

Remember
  • n² = n × n. Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.
  • n³ = n × n × n = n² × n. Cubes to 125: 1, 8, 27, 64, 125.
  • 1 and 64 are both square and cube numbers.
  • Gaps of 3, 5, 7, 9 … → think square numbers.
  • Position-to-term rules: square the position first, then add or subtract (10th term of position² + 3 = 103).
All pages in this course

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Number: integers and calculation

Number: place value, decimals, fractions, percentages and ratio

Geometry and measure

Statistics and probability

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