Does this 'trick' for calculating multiples of 11 always work? Why or why not?
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. . . .
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?
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.
Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?
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?
Can you explain the strategy for winning this game with any target?
The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?
This task encourages you to investigate the number of edging pieces and panes in different sized windows.
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
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.
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.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
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?
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?
Got It game for an adult and child. How can you play so that you know you will always win?
Strike it Out game for an adult and child. Can you stop your partner from being able to go?
This task follows on from Build it Up and takes the ideas into three dimensions!
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.
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.
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
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.
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
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?
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4
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?
Are these statements always true, sometimes true or never true?
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?
Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?
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 article for primary teachers discusses how we can help learners generalise and prove, using NRICH tasks as examples.
An investigation that gives you the opportunity to make and justify predictions.
Are these statements relating to odd and even numbers always true, sometimes true or never true?
This challenge asks you to imagine a snake coiling on itself.
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?
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.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?