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 continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
Find out what a "fault-free" rectangle is and try to make some of
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?
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
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?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
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?
Watch this film carefully. Can you find a general rule for
explaining when the dot will be this same distance from the
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Take a counter and surround it by a ring of other counters that
MUST touch two others. How many are needed?
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
Can you find all the ways to get 15 at the top of this triangle of numbers?
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?
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
This task follows on from Build it Up and takes the ideas into three dimensions!
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 make dice stairs using the rules stated? How do you know you have all the possible stairs?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
What happens when you round these three-digit numbers to the nearest 100?
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
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
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.
What happens when you round these numbers to the nearest whole number?
Delight your friends with this cunning trick! Can you explain how
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?
Find the sum of all three-digit numbers each of whose digits is
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 stations?
Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?
Strike it Out game for an adult and child. Can you stop your partner from being able to go?
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?
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?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
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.
An investigation that gives you the opportunity to make and justify
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
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?
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.
How many centimetres of rope will I need to make another mat just
like the one I have here?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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.