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.
Tom and Ben visited Numberland. Use the maps to work out the number
of points each of their routes scores.
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
Find the sum of all three-digit numbers each of whose digits is
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. . . .
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
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?
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?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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?
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.
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
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
This activity involves rounding four-digit numbers to the nearest thousand.
What happens when you round these three-digit numbers to the nearest 100?
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.
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?
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?
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?
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.
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.
Can you explain how this card trick works?
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.
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
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?
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?
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
Delight your friends with this cunning trick! Can you explain how
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
What happens when you round these numbers to the nearest whole number?
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?
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?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
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?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
This challenge asks you to imagine a snake coiling on itself.
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?
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.
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.