Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?
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?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
Find out what a "fault-free" rectangle is and try to make some of
Ben and his mum are planting garlic. Use the interactivity to help
you find out how many cloves of garlic they might have had.
Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?
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?
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?
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
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 is about finding the difference between numbers which have the same tens digit.
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.
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
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
In this problem we are looking at sets of parallel sticks that
cross each other. What is the least number of crossings you can
make? And the greatest?
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?
Find the sum of all three-digit numbers each of whose digits is
This is a game for two players. Can you find out how to be the first to get to 12 o'clock?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
Watch this film carefully. Can you find a general rule for
explaining when the dot will be this same distance from the
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
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
An investigation that gives you the opportunity to make and justify
This challenge asks you to imagine a snake coiling on itself.
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?
Tom and Ben visited Numberland. Use the maps to work out the number
of points each of their routes scores.
Can you dissect an equilateral triangle into 6 smaller ones? What
number of smaller equilateral triangles is it NOT possible to
dissect a larger equilateral triangle into?
For this challenge, you'll need to play Got It! Can you explain the
strategy for winning this game with any target?
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
This activity focuses on rounding to the nearest 10.
What happens if you join every second point on this circle? How
about every third point? Try with different steps and see if you
can predict what will happen.
In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?
How many different journeys could you make if you were going to
visit four stations in this network? How about if there were five
stations? Can you predict the number of journeys for seven
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?
This article for teachers describes several games, found on the
site, all of which have a related structure that can be used to
develop the skills of strategic planning.
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.
In this calculation, the box represents a missing digit. What could the digit be? What would the solution be in each case?
We can arrange dots in a similar way to the 5 on a dice and they
usually sit quite well into a rectangular shape. How many
altogether in this 3 by 5? What happens for other sizes?
What happens when you round these three-digit numbers to the nearest 100?
Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?
Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten
numbers from the bags above so that their total is 37.
What happens when you round these numbers to the nearest whole number?
It starts quite simple but great opportunities for number discoveries and patterns!