In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
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 film carefully. Can you find a general rule for
explaining when the dot will be this same distance from the
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?
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?
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?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
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?
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
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?
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?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
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 can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
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
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?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
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.
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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 two kinds of spirals for you to explore. What do you notice?
Find out what a "fault-free" rectangle is and try to make some of
How many centimetres of rope will I need to make another mat just
like the one I have here?
An investigation that gives you the opportunity to make and justify
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.
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?
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 ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
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?
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?
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?
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?
With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.
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?
Are these statements relating to odd and even numbers always true, sometimes true or never true?
This task follows on from Build it Up and takes the ideas into three dimensions!
Can you find all the ways to get 15 at the top of this triangle of numbers?