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There are **39** NRICH Mathematical resources connected to **Square numbers**, you may find related items under Properties of numbers.

Problem
Primary curriculum
Secondary curriculum
### Always, Sometimes or Never? Number

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

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Square Subtraction

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Generating Triples

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Light the Lights Again

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Sticky Numbers

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Picture a Pyramid ...

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Factors and Multiples Puzzle

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Up and Down Staircases

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Odd Squares

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Picturing Square Numbers

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Fitted

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cycling Squares

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Two Primes Make One Square

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

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### One Wasn't Square

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.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Fractions in a Box

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.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Iff

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

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Odd Differences

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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Making Boxes

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

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Swimming Pool Tiles

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

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Few and Far Between?

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Robert's Spreadsheet

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Four Coloured Lights

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Summing Squares

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Numbers as Shapes

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

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Triangles Within Squares

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Smith and Jones

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!

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### A Square Deal

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

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Graphing Number Patterns

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Seven Square Numbers

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.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Augustus' Age

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Special 24

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.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### A Right Charlie

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

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Triangular Triples

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Digat

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Square Routes

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Time of Birth

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Little Squares

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?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cuisenaire Squares

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

Age 7 to 11

Challenge Level