Square, Cube and Triangular Numbers in Reasoning
Square, cube and triangular numbers in reasoning covers recognising and using square numbers (n x n), cube numbers (n x n x n) and triangular numbers (the running total 1 + 2 + 3 + ... + n), and applying them within logic and pattern-based 11+ reasoning questions, such as identifying a number that matches several of these properties at once or continuing a pattern of growing shapes.
Before you start
Make sure you're comfortable with these topics first:
Method
- Learn the square numbers up to at least 15 squared (1, 4, 9, 16, 25, up to 225) so they can be recognised instantly.
- Learn the cube numbers up to at least 6 cubed (1, 8, 27, 64, 125, 216), since these appear less often but are still tested.
- Remember a triangular number is the total when you add the counting numbers up to n (1, 3, 6, 10, 15), and can be pictured as dots forming a triangle.
- To find the next triangular number, add the next counting number to the previous triangular number, rather than recalculating the whole sum each time.
- When a question mixes these number types, test each candidate against every clue in turn, rather than trying to solve all the clues at once.
- Where a diagram or growing pattern is given, count how the pattern grows from one step to the next and match that growth to a square, cube or triangular sequence.
Worked example
The triangular numbers are 1, 3, 6, 10, 15, 21. Find the 8th triangular number, and state whether it is also a square number.
- List the triangular numbers by adding the next counting number each time: 1, 3, 6, 10, 15, 21.
- Continue the pattern to the 7th triangular number: 21 + 7 = 28.
- Continue the pattern to the 8th triangular number: 28 + 8 = 36.
- Check whether 36 is a square number: 6 x 6 = 36, so yes, it is a square number.
- Final answer: the 8th triangular number is 36, and it is also a square number (6 squared).
Practice questions
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Q1What is 9 squared?Show answer
Answer: 81
Q2What is 4 cubed?Show answer
Answer: 64
Q3List the first six triangular numbers.Show answer
Answer: 1, 3, 6, 10, 15, 21
Q4Is 49 a square number, a cube number, both, or neither?Show answer
Answer: A square number (7 x 7 = 49)
Q5Find a number, other than 1, that is both a square number and a cube number, between 1 and 100.Show answer
Answer: 64 (8 squared = 4 cubed = 64)
Q6The 5th triangular number is 15. Work out the 6th triangular number.Show answer
Answer: 21 (15 + 6)
Q7Explain why triangular numbers can be found by adding consecutive counting numbers starting from 1.Show answer
Answer: Each triangular number represents a triangle-shaped arrangement of dots where every new row has exactly one more dot than the row before it, so the total after n rows is the sum of every counting number from 1 up to n.
Exam-style questions
Written in the style of a 11+ exam paper, with a full mark scheme.
List the square numbers between 1 and 100, then state which of them are also triangular numbers.
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A stack of oranges is arranged so that each layer is a smaller square than the one below: the bottom layer has 25 oranges arranged in a 5 by 5 square, the next layer up has 16, then 9, then 4, then 1 on top. (a) Explain why each layer's total is a square number. (b) Work out the total number of oranges in the whole stack.
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Free printable worksheet
Want more practice on paper? Download the square, cube and triangular numbers in reasoning worksheet pack - 5 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 23 of 11+ Maths Workbook 4, the whole course as one free printable PDF.
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