Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
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.
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
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
How many centimetres of rope will I need to make another mat just
like the one I have here?
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?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
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.
Are these statements always true, sometimes true or never true?
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .
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.
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Consider all two digit numbers (10, 11, . . . ,99). In writing down
all these numbers, which digits occur least often, and which occur
most often ? What about three digit numbers, four digit numbers. . . .
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
This challenge asks you to imagine a snake coiling on itself.
Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9,
12, 15... other squares? 8, 11, 14... other squares?
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
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
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?
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?
Here are two kinds of spirals for you to explore. What do you notice?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
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?
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
Find out what a "fault-free" rectangle is and try to make some of
An investigation that gives you the opportunity to make and justify
Benâ€™s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?
One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
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.
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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.
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
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?
The diagram shows a 5 by 5 geoboard with 25 pins set out in a square array. Squares are made by stretching rubber bands round specific pins. What is the total number of squares that can be made on a. . . .
Are these statements relating to odd and even numbers always true, sometimes true or never true?
Strike it Out game for an adult and child. Can you stop your partner from being able to go?
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?
Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?