Reasoning, Convincing and Proof is part of our Developing Mathematical Thinking collection.
problem
Favourite
Fruity Totals
Age
7 to 16
Challenge level
In this interactivity each fruit has a hidden value. Can you deduce what each one is worth?
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Missing Multipliers
Age
7 to 14
Challenge level
What is the smallest number of answers you need to reveal in order to work out the missing headers?
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The Number Jumbler
Age
7 to 14
Challenge level
The Number Jumbler can always work out your chosen symbol. Can you work out how?
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5 by 5 Mathdokus
Age
7 to 16
Challenge level
Can you use the clues to complete these 5 by 5 Mathematical Sudokus?
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More Less is More
Age
7 to 14
Challenge level
In each of these games, you will need a little bit of luck, and your knowledge of place value to develop a winning strategy.
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Subtraction Surprise
Age
7 to 14
Challenge level
Try out some calculations. Are you surprised by the results?
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Your number is...
Age
7 to 14
Challenge level
Think of a number and follow the machine's instructions... I know what your number is! Can you explain how I know?
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Statement Snap
Age
7 to 14
Challenge level
You'll need to know your number properties to win a game of Statement Snap...
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Number Lines in Disguise
Age
7 to 14
Challenge level
Some of the numbers have fallen off Becky's number line. Can you figure out what they were?
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Remainders
Age
7 to 14
Challenge level
I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?
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Triangles in circles
Age
11 to 14
Challenge level
Can you find triangles on a 9-point circle? Can you work out their angles?
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Odds and Evens
Age
11 to 14
Challenge level
Are these games fair? How can you tell?
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What numbers can we make?
Age
11 to 14
Challenge level
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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Semi-regular Tessellations
Age
11 to 16
Challenge level
Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?
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Place your orders
Age
11 to 14
Challenge level
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
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American Billions
Age
11 to 14
Challenge level
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...
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Olympic Measures
Age
11 to 14
Challenge level
These Olympic quantities have been jumbled up! Can you put them back together again?
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Strange Bank Account
Age
11 to 14
Challenge level
Imagine a very strange bank account where you are only allowed to do two things...
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Crossed Ends
Age
11 to 14
Challenge level
Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?
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Multiples Sudoku
Age
11 to 14
Challenge level
Each clue in this Sudoku is the product of the two numbers in adjacent cells.
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Special Numbers
Age
11 to 14
Challenge level
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?
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Multiple Surprises
Age
11 to 16
Challenge level
Sequences of multiples keep cropping up...
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Tilted Squares
Age
11 to 14
Challenge level
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
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Cyclic Quadrilaterals
Age
11 to 16
Challenge level
Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?
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Mixing Lemonade
Age
11 to 14
Challenge level
Can you work out which drink has the stronger flavour?
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Polygon Rings
Age
11 to 14
Challenge level
Join pentagons together edge to edge. Will they form a ring?
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Polygon Pictures
Age
11 to 14
Challenge level
Can you work out how these polygon pictures were drawn, and use that to figure out their angles?
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Gabriel's Problem
Age
11 to 14
Challenge level
Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?
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Calendar Capers
Age
11 to 14
Challenge level
Choose any three by three square of dates on a calendar page...
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Arithmagons
Age
11 to 16
Challenge level
Can you find the values at the vertices when you know the values on the edges?
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What does it all add up to?
Age
11 to 18
Challenge level
If you take four consecutive numbers and add them together, the answer will always be even. What else do you notice?
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Favourite
Fill Me Up
Age
11 to 14
Challenge level
Can you sketch graphs to show how the height of water changes in different containers as they are filled?
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Legs Eleven
Age
11 to 14
Challenge level
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?
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Favourite
Squares in rectangles
Age
11 to 14
Challenge level
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle
contains 20 squares. What size rectangle(s) contain(s) exactly 100
squares? Can you find them all?
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Favourite
Product Sudoku
Age
11 to 16
Challenge level
The clues for this Sudoku are the product of the numbers in adjacent squares.
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Triangle Numbers
Age
11 to 14
Challenge level
Take a look at the multiplication square. The first eleven triangle
numbers have been identified. Can you see a pattern? Does the
pattern continue?
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What numbers can we make now?
Age
11 to 14
Challenge level
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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All in a jumble
Age
11 to 14
Challenge level
My measurements have got all jumbled up! Swap them around and see
if you can find a combination where every measurement is valid.
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Thousands and Millions
Age
11 to 14
Challenge level
Here's a chance to work with large numbers...
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Marbles in a box
Age
11 to 16
Challenge level
How many winning lines can you make in a three-dimensional version of noughts and crosses?
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Think of Two Numbers
Age
11 to 14
Challenge level
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
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Power mad!
Age
11 to 14
Challenge level
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
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Impossibilities
Age
11 to 14
Challenge level
Just because a problem is impossible doesn't mean it's difficult...
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Overlaps
Age
11 to 14
Challenge level
Can you find ways to put numbers in the overlaps so the rings have equal totals?
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Reversals
Age
11 to 14
Challenge level
Where should you start, if you want to finish back where you started?
problem
Favourite
Take Three From Five
Age
11 to 16
Challenge level
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?
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Same length
Age
11 to 16
Challenge level
Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?
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Fibonacci Surprises
Age
11 to 14
Challenge level
Play around with the Fibonacci sequence and discover some surprising results!
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More Number Pyramids
Age
11 to 14
Challenge level
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
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Seven Squares - Group-worthy Task
Age
11 to 14
Challenge level
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?
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Pythagoras Proofs
Age
11 to 16
Challenge level
Can you make sense of these three proofs of Pythagoras' Theorem?
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Tower of Hanoi
Age
11 to 14
Challenge level
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
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Star Polygons
Age
11 to 14
Challenge level
Draw some stars and measure the angles at their points. Can you find and prove a result about their sum?
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Always a multiple?
Age
11 to 14
Challenge level
Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...
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Differences
Age
11 to 14
Challenge level
Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?
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Consecutive negative numbers
Age
11 to 14
Challenge level
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
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Shopping Basket
Age
11 to 16
Challenge level
The items in the shopping basket add and multiply to give the same amount. What could their prices be?
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the greedy algorithm
Age
11 to 14
Challenge level
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.
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Favourite
Generating Triples
Age
14 to 16
Challenge level
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?
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Favourite
Speed-time problems at the Olympics
Age
14 to 16
Challenge level
Have you ever wondered what it would be like to race against Usain Bolt?
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Finding factors
Age
14 to 16
Challenge level
Can you find the hidden factors which multiply together to produce each quadratic expression?
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Favourite
Factorising with Multilink
Age
14 to 16
Challenge level
Can you find out what is special about the dimensions of rectangles you can make with squares, sticks and units?
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Favourite
How old am I?
Age
14 to 16
Challenge level
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?
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Favourite
Nutrition and Cycling
Age
14 to 16
Challenge level
Andy wants to cycle from Land's End to John o'Groats. Will he be able to eat enough to keep him going?
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Double Trouble
Age
14 to 16
Challenge level
Simple additions can lead to intriguing results...
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Mega Quadratic Equations
Age
14 to 18
Challenge level
What do you get when you raise a quadratic to the power of a quadratic?
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Favourite
Curvy areas
Age
14 to 16
Challenge level
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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Speeding boats
Age
14 to 16
Challenge level
Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?
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Circles in quadrilaterals
Age
14 to 16
Challenge level
Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.
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A little light thinking
Age
14 to 16
Challenge level
Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?
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Salinon
Age
14 to 16
Challenge level
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?
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Common Divisor
Age
14 to 18
Challenge level
Can you find out what numbers divide these expressions? Can you prove that they are always divisors?
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Multiplication square
Age
14 to 16
Challenge level
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
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Picture Story
Age
14 to 16
Challenge level
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
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Doesn't add up
Age
14 to 16
Challenge level
In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?
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Multiplication arithmagons
Age
14 to 16
Challenge level
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
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Quad in Quad
Age
14 to 18
Challenge level
Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?
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Why 24?
Age
14 to 16
Challenge level
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.
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Iff
Age
14 to 18
Challenge level
Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?
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Kite in a Square
Age
14 to 18
Challenge level
Can you make sense of the three methods to work out what fraction of the total area is shaded?
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LCM Sudoku
Age
14 to 16
Challenge level
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
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Favourite
CD Heaven
Age
14 to 16
Challenge level
All CD Heaven stores were given the same number of a popular CD to
sell for £24. In their two week sale each store reduces the
price of the CD by 25% ... How many CDs did the store sell at each
price?
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Favourite
Pythagoras Perimeters
Age
14 to 16
Challenge level
If you know the perimeter of a right angled triangle, what can you say about the area?
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Repetitiously
Age
14 to 16
Challenge level
Can you express every recurring decimal as a fraction?
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Terminology
Age
14 to 16
Challenge level
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
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Favourite
Difference of Two Squares
Age
14 to 16
Challenge level
What is special about the difference between squares of numbers adjacent to multiples of three?
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Always Perfect
Age
14 to 18
Challenge level
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
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Back fitter
Age
14 to 18
Challenge level
10 graphs of experimental data are given. Can you use a spreadsheet to find algebraic graphs which match them closely, and thus discover the formulae most likely to govern the underlying processes?
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Favourite
Puzzling Place Value
Age
14 to 16
Challenge level
Can you explain what is going on in these puzzling number tricks?
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Favourite
Painted Cube
Age
14 to 16
Challenge level
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?
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Favourite
Partly Painted Cube
Age
14 to 16
Challenge level
Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?
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Favourite
Latin Numbers
Age
14 to 16
Challenge level
Can you create a Latin Square from multiples of a six digit number?
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Harmonic Triangle
Age
14 to 16
Challenge level
Can you see how to build a harmonic triangle? Can you work out the next two rows?
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2-Digit Square
Age
14 to 16
Challenge level
A 2-Digit number is squared. When this 2-digit number is reversed
and squared, the difference between the squares is also a square.
What is the 2-digit number?
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Favourite
Difference Sudoku
Age
14 to 16
Challenge level
Use the differences to find the solution to this Sudoku.
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Angle Trisection
Age
14 to 16
Challenge level
It is impossible to trisect an angle using only ruler and compasses
but it can be done using a carpenter's square.
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Parabella
Age
16 to 18
Challenge level
This is a beautiful result involving a parabola and parallels.
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Direct logic
Age
16 to 18
Challenge level
Can you work through these direct proofs, using our interactive proof sorters?
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Integration matcher
Age
16 to 18
Challenge level
Can you match the charts of these functions to the charts of their integrals?