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?
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?
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 ...?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
Tom and Ben visited Numberland. Use the maps to work out the number
of points each of their routes scores.
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 see how to build a harmonic triangle? Can you work out the next two rows?
An investigation that gives you the opportunity to make and justify
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
This challenge asks you to imagine a snake coiling on itself.
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?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
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?
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?
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.
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
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?
It would be nice to have a strategy for disentangling any tangled
Can you tangle yourself up and reach any fraction?
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.
Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9,
12, 15... other squares? 8, 11, 14... other squares?
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.
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
The Egyptians expressed all fractions as the sum of different unit
fractions. Here is a chance to explore how they could have written
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 numbers from a sequence. What numbers can we make now?
What happens when you round these numbers to the nearest whole number?
What happens when you round these three-digit numbers to the nearest 100?
Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
Find out what a "fault-free" rectangle is and try to make some of
How many centimetres of rope will I need to make another mat just
like the one I have here?
Can all unit fractions be written as the sum of two unit fractions?
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
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.
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. . . .
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?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
Take any whole number between 1 and 999, add the squares of the
digits to get a new number. Make some conjectures about what
happens in general.
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?