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?
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.
Watch this film carefully. Can you find a general rule for
explaining when the dot will be this same distance from the
Take a counter and surround it by a ring of other counters that
MUST touch two others. How many are needed?
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?
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?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
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?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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?
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 could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
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.
Are these statements relating to odd and even numbers always true, sometimes true or never true?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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 the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
Delight your friends with this cunning trick! Can you explain how
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.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
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
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.
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
Imagine a large cube made from small red cubes being dropped into a
pot of yellow paint. How many of the small cubes will have yellow
paint on their faces?
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?
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'.
Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9,
12, 15... other squares? 8, 11, 14... other squares?
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.
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?
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
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?
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
Explore the effect of reflecting in two intersecting mirror lines.
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
Are these statements always true, sometimes true or never true?
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?
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?