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.
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.
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
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?
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
Got It game for an adult and child. How can you play so that you know you will always win?
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.
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
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
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?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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?
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
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. . . .
The number of plants in Mr McGregor's magic potting shed increases
overnight. He'd like to put the same number of plants in each of
his gardens, planting one garden each day. How can he do it?
This activity involves rounding four-digit numbers to the nearest thousand.
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?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
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?
Find the sum of all three-digit numbers each of whose digits is
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?
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
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?
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?
An investigation that gives you the opportunity to make and justify
Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?
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. . . .
Tom and Ben visited Numberland. Use the maps to work out the number
of points each of their routes scores.
What happens when you round these three-digit numbers to the nearest 100?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
A collection of games on the NIM theme
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
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?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
Start with any number of counters in any number of piles. 2 players
take it in turns to remove any number of counters from a single
pile. The winner is the player to take the last counter.
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.
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.
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
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.
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
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.