In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?
Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?
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 see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?
Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?
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?
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
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.
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.
Can you work out how to win this game of Nim? Does it matter if you go first or second?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
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.
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.
Find out what a "fault-free" rectangle is and try to make some of your own.
Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?
Delight your friends with this cunning trick! Can you explain how it works?
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.
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 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?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
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?
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?
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
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.
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
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
Here are two kinds of spirals for you to explore. What do you notice?
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?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
A collection of games on the NIM theme
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.
Can you find sets of sloping lines that enclose a square?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
How many centimetres of rope will I need to make another mat just like the one I have here?
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.