This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
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?
Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.
While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
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 happens?
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?
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?
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?
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?
An investigation that gives you the opportunity to make and justify predictions.
This task follows on from Build it Up and takes the ideas into three dimensions!
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
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.
Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
Surprise your friends with this magic square trick.
In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
What happens when you round these three-digit numbers to the nearest 100?
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.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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?
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?
This challenge asks you to imagine a snake coiling on itself.
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
It starts quite simple but great opportunities for number discoveries and patterns!
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
Try out this number trick. What happens with different starting numbers? What do you notice?
A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.
These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?
Does this 'trick' for calculating multiples of 11 always work? Why or why not?
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.
Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.
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 stations?
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?
Find the sum of all three-digit numbers each of whose digits is odd.
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
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. . . .
Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?
This task encourages you to investigate the number of edging pieces and panes in different sized windows.
Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?
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?