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?
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?
Watch this animation. What do you see? Can you explain why this happens?
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 find a way of counting the spheres in these arrangements?
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?
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!
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
Delight your friends with this cunning trick! Can you explain how it works?
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 continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
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.
Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?
How many centimetres of rope will I need to make another mat just like the one I have here?
Can you describe this route to infinity? Where will the arrows take you next?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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.
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.
Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.
This task follows on from Build it Up and takes the ideas into three dimensions!
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
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?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?
Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?
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?
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?
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?
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.
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
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 article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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?
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
Does this 'trick' for calculating multiples of 11 always work? Why or why not?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
Take a look at the video of this trick. Can you perform it yourself? Why is this maths and not magic?
Are these statements always true, sometimes true or never true?
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.