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?
Watch this animation. What do you see? Can you explain why this happens?
Can you find a way of counting the spheres in these arrangements?
Delight your friends with this cunning trick! Can you explain how it works?
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
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?
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?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
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.
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
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?
Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
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?
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 the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
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?
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?
Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
These tasks give learners chance to generalise, which involves identifying an underlying structure.
Strike it Out game for an adult and child. Can you stop your partner from being able to go?
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?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Got It game for an adult and child. How can you play so that you know you will always win?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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.
Can you describe this route to infinity? Where will the arrows take you next?
This task encourages you to investigate the number of edging pieces and panes in different sized windows.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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.
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.
A collection of games on the NIM theme
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
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?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
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
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.