Square numbers

There are 44 NRICH Mathematical resources connected to Square numbers

problem

Favourite

Square subtraction

Age

7 to 11

Challenge level

Look at what happens when you take a number, square it and subtract your answer. What kind of number do you get? Can you prove it?

problem

Favourite

Filling the gaps

Age

14 to 16

Challenge level

Which numbers can we write as a sum of square numbers?

problem

Favourite

Generating triples

Age

14 to 16

Challenge level

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

problem

Favourite

Fitted

Age

7 to 11

Challenge level

Nine squares with side lengths 1, 4, 7, 8, 9, 10, 14, 15, and 18 cm can be fitted together to form a rectangle. What are the dimensions of the rectangle?

problem

Favourite

Odd differences

Age

14 to 16

Challenge level

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = nÂ² Use the diagram to show that any odd number is the difference of two squares.

problem

Favourite

Picturing square numbers

Age

11 to 14

Challenge level

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

problem

Favourite

Odd squares

Age

7 to 11

Challenge level

Think of a number, square it and subtract your starting number. Is the number you're left with odd or even? How do the images help to explain this?

problem

Favourite

Up and down staircases

Age

7 to 11

Challenge level

One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?

problem

Favourite

Iff

Age

14 to 18

Challenge level

Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?

problem

Favourite

Factors and multiples puzzle

Age

11 to 14

Challenge level

Using your knowledge of the properties of numbers, can you fill all the squares on the board?

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For younger learners

Square numbers

Square subtraction

Filling the gaps

Generating triples

Fitted

Odd differences

Picturing square numbers

Odd squares

Up and down staircases

Iff

Factors and multiples puzzle