Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?
A country has decided to have just two different coins, 3z and 5z
coins. Which totals can be made? Is there a largest total that
cannot be made? How do you know?
Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .
Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?
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.
It starts quite simple but great opportunities for number discoveries and patterns!
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. . . .
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
Can you find the values at the vertices when you know the values on
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Charlie has moved between countries and the average income of both
has increased. How can this be so?
Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten.
Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .
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.
Can you explain how this card trick works?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?
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?
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?
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
We can show that (x + 1)² = x² + 2x + 1 by considering
the area of an (x + 1) by (x + 1) square. Show in a similar way
that (x + 2)² = x² + 4x + 4
This task follows on from Build it Up and takes the ideas into three dimensions!
A package contains a set of resources designed to develop
pupils’ mathematical thinking. This package places a
particular emphasis on “generalising” and is designed
to meet the. . . .
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
Here are two kinds of spirals for you to explore. What do you notice?
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
How many centimetres of rope will I need to make another mat just
like the one I have here?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
An investigation that gives you the opportunity to make and justify
Find out what a "fault-free" rectangle is and try to make some of
Can you see how to build a harmonic triangle? Can you work out the next two rows?
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 tangle yourself up and reach any fraction?
It would be nice to have a strategy for disentangling any tangled
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
Jo has three numbers which she adds together in pairs. When she
does this she has three different totals: 11, 17 and 22 What are
the three numbers Jo had to start with?”
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?
Are these statements always true, sometimes true or never true?
Can you find all the ways to get 15 at the top of this triangle of 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?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
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?
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?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
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
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?