There are 36 NRICH Mathematical resources connected to Square numbers, you may find related items under Properties of Numbers.
Broad Topics > Properties of Numbers > Square numbers
Always, Sometimes or Never? Number
Age 7 to 11
Challenge Level
Are these statements always true, sometimes true or never true?
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?
Light the Lights Again
Age 7 to 11
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?
Sticky Numbers
Age 11 to 14
Challenge Level
Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?
Picture a Pyramid ...
Age 7 to 11
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?
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?
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?
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?
Cycling Squares
Age 7 to 11
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?
Two Primes Make One Square
Age 7 to 11
Challenge Level
Can you make square numbers by adding two prime numbers together?
One Wasn't Square
Age 7 to 11
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.
Fractions in a Box
Age 7 to 11
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.
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?
Odd Differences
Age 14 to 16
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.
Making Boxes
Age 7 to 11
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?
Swimming Pool Tiles
Age 7 to 11
Challenge Level
This activity creates an opportunity to explore all kinds of number-related patterns.
Few and Far Between?
Age 16 to 18
Challenge Level
Can you find some Pythagorean Triples where the two smaller numbers differ by 1?
Robert's Spreadsheet
Age 14 to 16
Challenge Level
Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?
Four Coloured Lights
Age 11 to 14
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?
Summing Squares
Age 14 to 16
Challenge Level
Discover a way to sum square numbers by building cuboids from small cubes. Can you picture how the sequence will grow?
Numbers as Shapes
Age 5 to 7
Challenge Level
Use cubes to continue making the numbers from 7 to 20. Are they sticks, rectangles or squares?
Triangles Within Squares
Age 14 to 16
Challenge Level
Can you find a rule which relates triangular numbers to square numbers?
Smith and Jones
Age 14 to 16
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!
A Square Deal
Age 7 to 11
Challenge Level
Complete the magic square using the numbers 1 to 25 once each. Each row, column and diagonal adds up to 65.
Graphing Number Patterns
Age 7 to 11
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?
Seven Square Numbers
Age 7 to 11
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.
Augustus' Age
Age 7 to 11
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?
Special 24
Age 7 to 11
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.
A Right Charlie
Age 7 to 11
Challenge Level
Can you use this information to work out Charlie's house number?
Triangular Triples
Age 14 to 16
Challenge Level
Show that 8778, 10296 and 13530 are three triangular numbers and that they form a Pythagorean triple.
Digat
Age 11 to 14
Challenge Level
What is the value of the digit A in the sum below: [3(230 + A)]^2 = 49280A
Square Routes
Age 11 to 14
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?
Time of Birth
Age 11 to 14
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?
Little Squares
Age 5 to 7
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?
Cuisenaire Squares
Age 7 to 11
Challenge Level
These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?
NRICH is part of the family of activities in the Millennium Mathematics Project.