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?
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
Take a counter and surround it by a ring of other counters that
MUST touch two others. How many are needed?
Watch this film carefully. Can you find a general rule for
explaining when the dot will be this same distance from the
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
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 many moves does it take to swap over some red and blue frogs? Do you have a method?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
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?
Delight your friends with this cunning trick! Can you explain how
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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.
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?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
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?
Triangle numbers can be represented by a triangular array of
squares. What do you notice about the sum of identical triangle
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
Can you find the values at the vertices when you know the values on
Find out what a "fault-free" rectangle is and try to make some of
Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Imagine you have a large supply of 3kg and 8kg weights. How many of
each weight would you need for the average (mean) of the weights to
be 6kg? What other averages could you have?
Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?
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
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?
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?
Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9,
12, 15... other squares? 8, 11, 14... other squares?
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 explain how this card trick works?
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.
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
A country has decided to have just two different coins, 3z and 5z
coins. Which totals can be made? Is there a largest total that
cannot be made? How do you know?
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.
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Explore the effect of reflecting in two parallel mirror lines.
Explore the effect of combining enlargements.
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.
Explore the effect of reflecting in two intersecting mirror lines.
This challenge asks you to imagine a snake coiling on itself.
Find the sum of all three-digit numbers each of whose digits is
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?
Start with two numbers. This is the start of a sequence. The next
number is the average of the last two numbers. Continue the
sequence. What will happen if you carry on for ever?
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
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?