There are **36** NRICH Mathematical resources connected to **Square numbers**, you may find related items under Properties of Numbers.

Challenge Level

Are these statements always true, sometimes true or never true?

Challenge Level

Each light in this interactivity turns on according to a rule. What happens when you enter different numbers? Can you find the smallest number that lights up all four lights?

Challenge Level

Imagine a pyramid which is built in square layers of small cubes. If we number the cubes from the top, starting with 1, can you picture which cubes are directly below this first cube?

Challenge Level

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

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?

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?

Challenge Level

Can you make a cycle of pairs that add to make a square number using all the numbers in the box below, once and once only?

Challenge Level

Can you make square numbers by adding two prime numbers together?

Challenge Level

Mrs Morgan, the class's teacher, pinned numbers onto the backs of three children. Use the information to find out what the three numbers were.

Challenge Level

The discs for this game are kept in a flat square box with a square hole for each. Use the information to find out how many discs of each colour there are in the box.

Challenge Level

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

Challenge Level

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

Challenge Level

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

Challenge Level

This activity creates an opportunity to explore all kinds of number-related patterns.

Challenge Level

Can you find some Pythagorean Triples where the two smaller numbers differ by 1?

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?

Challenge Level

Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?

Challenge Level

Imagine a machine with four coloured lights which respond to different rules. Can you find the smallest possible number which will make all four colours light up?

Challenge Level

Discover a way to sum square numbers by building cuboids from small cubes. Can you picture how the sequence will grow?

Challenge Level

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

Challenge Level

Use cubes to continue making the numbers from 7 to 20. Are they sticks, rectangles or squares?

Challenge Level

Can you find a rule which relates triangular numbers to square numbers?

Challenge Level

Mr Smith and Mr Jones are two maths teachers. By asking questions, the answers to which may be right or wrong, Mr Jones is able to find the number of the house Mr Smith lives in... Or not!

Challenge Level

Complete the magic square using the numbers 1 to 25 once each. Each row, column and diagonal adds up to 65.

Challenge Level

Does a graph of the triangular numbers cross a graph of the six times table? If so, where? Will a graph of the square numbers cross the times table too?

Challenge Level

Add the sum of the squares of four numbers between 10 and 20 to the sum of the squares of three numbers less than 6 to make the square of another, larger, number.

Challenge Level

In 1871 a mathematician called Augustus De Morgan died. De Morgan made a puzzling statement about his age. Can you discover which year De Morgan was born in?

Challenge Level

Find another number that is one short of a square number and when you double it and add 1, the result is also a square number.

Challenge Level

Can you use this information to work out Charlie's house number?

Challenge Level

Show that 8778, 10296 and 13530 are three triangular numbers and that they form a Pythagorean triple.

Challenge Level

What is the value of the digit A in the sum below: [3(230 + A)]^2 = 49280A

Challenge Level

How many four digit square numbers are composed of even numerals? What four digit square numbers can be reversed and become the square of another number?

Challenge Level

A woman was born in a year that was a square number, lived a square number of years and died in a year that was also a square number. When was she born?

Challenge Level

Look at the squares in this problem. What does the next square look like? I draw a square with 81 little squares inside it. How long and how wide is my square?

Challenge Level

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?