Filter by: Content type: ALL Problems Articles Games Stage: All Stage 1&2 Stage 2&3 Stage 3&4 Stage 4&5 Challenge level:
These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?
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?
Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?
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?
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
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 your own.
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
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?
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?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.
Delight your friends with this cunning trick! Can you explain how it works?
Find the sum of all three-digit numbers each of whose digits is odd.
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?
What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
It starts quite simple but great opportunities for number discoveries and patterns!
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?
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.
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
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.
This challenge asks you to imagine a snake coiling on itself.
Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?
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.
Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?
If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
An investigation that gives you the opportunity to make and justify predictions.
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
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 moves.
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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?
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Start with two numbers. This is the start of a sequence. The next number is the average of the last two numbers. Continue the sequence. What will happen if you carry on for ever?
Explore the effect of reflecting in two parallel mirror lines.
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.
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?