Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
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 continue this pattern of triangles and begin to predict how
many sticks are used for each new "layer"?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
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?
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?
An investigation that gives you the opportunity to make and justify
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?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
Try adding together the dates of all the days in one week. Now
multiply the first date by 7 and add 21. Can you explain what
Find out what a "fault-free" rectangle is and try to make some of
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?
Find the sum of all three-digit numbers each of whose digits is
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.
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
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
Tom and Ben visited Numberland. Use the maps to work out the number
of points each of their routes scores.
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
What happens if you join every second point on this circle? How
about every third point? Try with different steps and see if you
can predict what will happen.
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.
Delight your friends with this cunning trick! Can you explain how
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
How many centimetres of rope will I need to make another mat just
like the one I have here?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
Take a counter and surround it by a ring of other counters that
MUST touch two others. How many are needed?
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?
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?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Ben’s class were making cutting up number tracks. First they
cut them into twos and added up the numbers on each piece. What
patterns could they see?
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?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Can you explain how this card trick works?
In each of the pictures the invitation is for you to: Count what
you see. Identify how you think the pattern would continue.
Can you work out how to win this game of Nim? Does it matter if you
go first or second?
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
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?
This challenge asks you to imagine a snake coiling on itself.
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?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Watch this film carefully. Can you find a general rule for
explaining when the dot will be this same distance from the
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?
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?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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?
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?