Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.
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?
You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .
List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?
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.
Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
Does this 'trick' for calculating multiples of 11 always work? Why or why not?
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
Can you explain the strategy for winning this game with any target?
The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
Got It game for an adult and child. How can you play so that you know you will always win?
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.
Strike it Out game for an adult and child. Can you stop your partner from being able to go?
Are these statements always true, sometimes true or never true?
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?
Investigate the different ways that fifteen schools could have given money in a charity fundraiser.
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?
Can you find the values at the vertices when you know the values on the edges?
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?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
An investigation that gives you the opportunity to make and justify predictions.
Ben’s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?
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?
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.
Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
These tasks give learners chance to generalise, which involves identifying an underlying structure.
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
This task follows on from Build it Up and takes the ideas into three dimensions!
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
This article for primary teachers discusses how we can help learners generalise and prove, using NRICH tasks as examples.
Are these statements relating to odd and even numbers always true, sometimes true or never true?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
Nim-7 game for an adult and child. Who will be the one to take the last counter?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Try out this number trick. What happens with different starting numbers? What do you notice?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
This challenge asks you to imagine a snake coiling on itself.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?