Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
Are these statements always true, sometimes true or never true?
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?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
Find some examples of pairs of numbers such that their sum is a
factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and
16 is a factor of 48.
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 pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Ben’s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?
Got It game for an adult and child. How can you play so that you know you will always win?
Choose any 3 digits and make a 6 digit number by repeating the 3
digits in the same order (e.g. 594594). Explain why whatever digits
you choose the number will always be divisible by 7, 11 and 13.
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.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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
List any 3 numbers. It is always possible to find a subset of
adjacent numbers that add up to a multiple of 3. Can you explain
why and prove it?
An investigation that gives you the opportunity to make and justify
This task follows on from Build it Up and takes the ideas into three dimensions!
This challenge asks you to imagine a snake coiling on itself.
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?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
Can you find all the ways to get 15 at the top of this triangle of numbers?
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Here are two kinds of spirals for you to explore. What do you notice?
This activity involves rounding four-digit numbers to the nearest thousand.
Find the sum of all three-digit numbers each of whose digits is
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
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?
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?
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?
Are these statements relating to odd and even numbers always true, sometimes true or never true?
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
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?
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.
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
Consider all two digit numbers (10, 11, . . . ,99). In writing down
all these numbers, which digits occur least often, and which occur
most often ? What about three digit numbers, four digit numbers. . . .
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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 many centimetres of rope will I need to make another mat just
like the one I have here?
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?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.
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 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?