The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
Here are two kinds of spirals for you to explore. What do you notice?
Are these statements relating to calculation and properties of shapes always true, sometimes true or never true?
Are these statements always true, sometimes true or never true?
If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice?
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.
This challenge asks you to imagine a snake coiling on itself.
It starts quite simple but great opportunities for number discoveries and patterns!
Two children made up a game as they walked along the garden paths. Can you find out their scores? Can you find some paths of your own?
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
The aim of the game is to slide the green square from the top right
hand corner to the bottom left hand corner in the least number of
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
How many centimetres of rope will I need to make another mat just
like the one I have here?
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
This activity involves rounding four-digit numbers to the nearest thousand.
This problem challenges you to find out how many odd numbers there are between pairs of numbers. Can you find a pair of numbers that has four odds between them?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
This activity focuses on rounding to the nearest 10.
Are these statements relating to odd and even numbers always true, sometimes true or never true?
Stop the Clock game for an adult and child. How can you make sure you always win this game?
Strike it Out game for an adult and child. Can you stop your partner from being able to go?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
Can you find all the ways to get 15 at the top of this triangle of numbers?
This task follows on from Build it Up and takes the ideas into three dimensions!
Got It game for an adult and child. How can you play so that you know you will always win?
Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?
Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?
What happens when you round these three-digit numbers to the nearest 100?
An investigation that gives you the opportunity to make and justify
In this calculation, the box represents a missing digit. What could the digit be? What would the solution be in each case?
This challenge is about finding the difference between numbers which have the same tens digit.
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
What happens when you round these numbers to the nearest whole number?
Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
How can you arrange these 10 matches in four piles so that when you
move one match from three of the piles into the fourth, you end up
with the same arrangement?
Watch this film carefully. Can you find a general rule for
explaining when the dot will be this same distance from the
Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the
numbers on each circle add up to the same amount. Can you find the
rule for giving another set of six numbers?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
Try adding together the dates of all the days in one week. Now
multiply the first date by 7 and add 21. Can you explain what