Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
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?
For this challenge, you'll need to play Got It! Can you explain the
strategy for winning this game with any target?
An investigation that gives you the opportunity to make and justify
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.
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
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
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?
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?
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.
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?
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.
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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?
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.
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
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?
Tom and Ben visited Numberland. Use the maps to work out the number
of points each of their routes scores.
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
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?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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?
Delight your friends with this cunning trick! Can you explain how
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
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
This challenge asks you to imagine a snake coiling on itself.
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.
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?
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?
Can you explain how this card trick works?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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.
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. . . .
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
Find the sum of all three-digit numbers each of whose digits is
Can you work out how to win this game of Nim? Does it matter if you
go first or second?
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?
How many pairs of numbers can you find that add up to a multiple of
11? Do you notice anything interesting about your results?
Can you find the values at the vertices when you know the values on
Can you describe this route to infinity? Where will the arrows take you next?
Can you tangle yourself up and reach any fraction?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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?
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?
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.