Filter by: Content type: ALL Problems Articles Games Stage: All Stage 1&2 Stage 2&3 Stage 3&4 Stage 4&5 Challenge level:
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?
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?
These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?
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?
Delight your friends with this cunning trick! Can you explain how it works?
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?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
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.
Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
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?
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?
Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.
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.
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
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.
Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?
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?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
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.
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
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?
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?
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.
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.
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?
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?
Find the sum of all three-digit numbers each of whose digits is odd.
Can you explain how this card trick works?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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?
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 continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
This challenge asks you to imagine a snake coiling on itself.
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?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The winner is the player to take the last counter.
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.
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?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?