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?
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?
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?
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?
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 see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
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?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
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.
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?
Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
Find out what a "fault-free" rectangle is and try to make some of your own.
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?
Here are two kinds of spirals for you to explore. What do you notice?
This challenge asks you to imagine a snake coiling on itself.
Can you explain how this card trick works?
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
Can you describe this route to infinity? Where will the arrows take you next?
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
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 find an efficient method to work out how many handshakes there would be if hundreds of people met?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?
Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?
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.
A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.
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?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?
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?
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?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Find the sum of all three-digit numbers each of whose digits is odd.
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?
Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?
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?
Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.
Can you find all the ways to get 15 at the top of this triangle of numbers?
This task follows on from Build it Up and takes the ideas into three dimensions!
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?