Filter by: Content type: ALL Problems Articles Games Stage: All Stage 1&2 Stage 2&3 Stage 3&4 Stage 4&5 Challenge level:
Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?
Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?
Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?
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?
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 ...?
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?
Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
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?
Delight your friends with this cunning trick! Can you explain how it works?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
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?
Find the sum of all three-digit numbers each of whose digits is odd.
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
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?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.
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.
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.
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?
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.
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
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.
An investigation that gives you the opportunity to make and justify predictions.
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
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.
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.
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Can you explain how this card trick works?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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?
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?
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.
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?
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.
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?
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?
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?
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.