Or search by topic
Reasoning, Convincing and Proof is part of our Developing Mathematical Thinking collection.
What is the smallest number of answers you need to reveal in order to work out the missing headers?
The Number Jumbler can always work out your chosen symbol. Can you work out how?
Think of a number and follow the machine's instructions... I know what your number is! Can you explain how I know?
In each of these games, you will need a little bit of luck, and your knowledge of place value to develop a winning strategy.
Try out some calculations. Are you surprised by the results?
You'll need to know your number properties to win a game of Statement Snap...
I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?
Can you use the clues to complete these 5 by 5 Mathematical Sudokus?
In this interactivity each fruit has a hidden value. Can you deduce what each one is worth?
Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...
Each clue in this Sudoku is the product of the two numbers in adjacent cells.
Choose any three by three square of dates on a calendar page...
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?
These Olympic quantities have been jumbled up! Can you put them back together again?
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
Imagine a very strange bank account where you are only allowed to do two things...
Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?
15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?
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.
Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11?
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
Play around with the Fibonacci sequence and discover some surprising results!
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Just because a problem is impossible doesn't mean it's difficult...
Can you find ways to put numbers in the overlaps so the rings have equal totals?
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?
Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?
Where should you start, if you want to finish back where you started?
Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...
My measurements have got all jumbled up! Swap them around and see if you can find a combination where every measurement is valid.
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
The Egyptians expressed all fractions as the sum of different unit fractions. The Greedy Algorithm might provide us with an efficient way of doing this.
A collection of short Stage 3 and 4 problems requiring Reasoning, Convincing and Proving.
Can you find the values at the vertices when you know the values on the edges?
There are lots of different methods to find out what the shapes are worth - how many can you find?
Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?
The clues for this Sudoku are the product of the numbers in adjacent squares.
How many winning lines can you make in a three-dimensional version of noughts and crosses?
Caroline and James pick sets of five numbers. Charlie tries to find three that add together to make a multiple of three. Can they stop him?
Can you make sense of these three proofs of Pythagoras' Theorem?
Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?
The items in the shopping basket add and multiply to give the same amount. What could their prices be?
If you take four consecutive numbers and add them together, the answer will always be even. What else do you notice?
This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Can you find the hidden factors which multiply together to produce each quadratic expression?
Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?
In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?
Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?
Can you find out what is special about the dimensions of rectangles you can make with squares, sticks and units?
Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?
Andy wants to cycle from Land's End to John o'Groats. Will he be able to eat enough to keep him going?
Have you ever wondered what it would be like to race against Usain Bolt?
The sums of the squares of three related numbers is also a perfect square - can you explain why?
Can you explain what is going on in these puzzling number tricks?
Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
What is special about the difference between squares of numbers adjacent to multiples of three?
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?
Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
If you know the perimeter of a right angled triangle, what can you say about the area?
Use the differences to find the solution to this Sudoku.
It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.
Can you create a Latin Square from multiples of a six digit number?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?
Can you make sense of the three methods to work out what fraction of the total area is shaded?
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?
Can you match the charts of these functions to the charts of their integrals?
Can you work through these direct proofs, using our interactive proof sorters?
Do you have enough information to work out the area of the shaded quadrilateral?
Sort these mathematical propositions into a series of 8 correct statements.
Which of these triangular jigsaws are impossible to finish?