List

Reasoning, Convincing and Proving - Secondary Students

Statement Snap
problem
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Statement Snap

Age
7 to 14
Challenge level
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You'll need to know your number properties to win a game of Statement Snap...
Fruity Totals
problem
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Fruity Totals

Age
7 to 16
Challenge level
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In this interactivity each fruit has a hidden value. Can you deduce what each one is worth?

missing multipliers
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Missing Multipliers

Age
7 to 14
Challenge level
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What is the smallest number of answers you need to reveal in order to work out the missing headers?
The Number Jumbler
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The Number Jumbler

Age
7 to 14
Challenge level
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The Number Jumbler can always work out your chosen symbol. Can you work out how?
5 by 5 Mathdokus
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5 by 5 Mathdokus

Age
7 to 16
Challenge level
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Can you use the clues to complete these 5 by 5 Mathematical Sudokus?
More Less is More
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More Less is More

Age
7 to 14
Challenge level
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In each of these games, you will need a little bit of luck, and your knowledge of place value to develop a winning strategy.
Subtraction Surprise
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Subtraction Surprise

Age
7 to 14
Challenge level
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Try out some calculations. Are you surprised by the results?
Your number is...
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Your number is...

Age
7 to 14
Challenge level
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Think of a number and follow the machine's instructions... I know what your number is! Can you explain how I know?
Number Lines in Disguise
problem
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Number Lines in Disguise

Age
7 to 14
Challenge level
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Some of the numbers have fallen off Becky's number line. Can you figure out what they were?
Remainders
problem
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Remainders

Age
7 to 14
Challenge level
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I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?
Gabriel's Problem
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Gabriel's Problem

Age
11 to 14
Challenge level
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Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?
Odds and Evens
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Odds and Evens

Age
11 to 14
Challenge level
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Are these games fair? How can you tell?
Semi-regular Tessellations
problem
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Semi-regular Tessellations

Age
11 to 16
Challenge level
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Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

What numbers can we make?
problem
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What numbers can we make?

Age
11 to 14
Challenge level
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Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

Place your orders
problem
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Place your orders

Age
11 to 14
Challenge level
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Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
American Billions
problem
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American Billions

Age
11 to 14
Challenge level
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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...
Crossed Ends
problem
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Crossed Ends

Age
11 to 14
Challenge level
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Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?
Olympic Measures
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Olympic Measures

Age
11 to 14
Challenge level
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These Olympic quantities have been jumbled up! Can you put them back together again?
Strange Bank Account
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Strange Bank Account

Age
11 to 14
Challenge level
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Imagine a very strange bank account where you are only allowed to do two things...

Multiples Sudoku
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Multiples Sudoku

Age
11 to 14
Challenge level
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Each clue in this Sudoku is the product of the two numbers in adjacent cells.
Cryptarithms
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Cryptarithms

Age
11 to 14
Challenge level
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Can you crack these cryptarithms?
Special Numbers
problem
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Special Numbers

Age
11 to 14
Challenge level
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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?

Tilted Squares
problem
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Tilted Squares

Age
11 to 14
Challenge level
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It's easy to work out the areas of most squares that we meet, but what if they were tilted?

Cyclic Quadrilaterals
problem
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Cyclic Quadrilaterals

Age
11 to 16
Challenge level
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Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

Multiple Surprises
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Multiple Surprises

Age
11 to 16
Challenge level
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Sequences of multiples keep cropping up...
Mixing Lemonade
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Mixing Lemonade

Age
11 to 14
Challenge level
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Can you work out which drink has the stronger flavour?

Polygon Rings
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Polygon Rings

Age
11 to 14
Challenge level
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Join pentagons together edge to edge. Will they form a ring?

Calendar Capers
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Calendar Capers

Age
11 to 14
Challenge level
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Choose any three by three square of dates on a calendar page...
Arithmagons
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Arithmagons

Age
11 to 16
Challenge level
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Can you find the values at the vertices when you know the values on the edges?

Polygon Pictures
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Polygon Pictures

Age
11 to 14
Challenge level
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Can you work out how these polygon pictures were drawn, and use that to figure out their angles?

Triangles in circles
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Triangles in circles

Age
11 to 14
Challenge level
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Can you find triangles on a 9-point circle? Can you work out their angles?

Always a multiple?
problem
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Always a multiple?

Age
11 to 14
Challenge level
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Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...
Legs Eleven
problem
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Legs Eleven

Age
11 to 14
Challenge level
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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?
Squares in rectangles
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Squares in rectangles

Age
11 to 14
Challenge level
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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?
Product Sudoku
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Product Sudoku

Age
11 to 16
Challenge level
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The clues for this Sudoku are the product of the numbers in adjacent squares.

What does it all add up to?
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What does it all add up to?

Age
11 to 18
Challenge level
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If you take four consecutive numbers and add them together, the answer will always be even. What else do you notice?
Triangle Numbers
problem
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Triangle Numbers

Age
11 to 14
Challenge level
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Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?
Fill Me Up
problem
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Fill Me Up

Age
11 to 14
Challenge level
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Can you sketch graphs to show how the height of water changes in different containers as they are filled?

All in a jumble
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All in a jumble

Age
11 to 14
Challenge level
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My measurements have got all jumbled up! Swap them around and see if you can find a combination where every measurement is valid.
Thousands and Millions
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Thousands and Millions

Age
11 to 14
Challenge level
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Here's a chance to work with large numbers...
What numbers can we make now?
problem
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What numbers can we make now?

Age
11 to 14
Challenge level
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Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

Marbles in a box
problem
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Marbles in a box

Age
11 to 16
Challenge level
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How many winning lines can you make in a three-dimensional version of noughts and crosses?
Think of Two Numbers
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Think of Two Numbers

Age
11 to 14
Challenge level
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Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
Power mad!
problem
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Power mad!

Age
11 to 14
Challenge level
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Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

Take Three From Five
problem
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Take Three From Five

Age
11 to 16
Challenge level
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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?

Impossibilities
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Impossibilities

Age
11 to 14
Challenge level
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Just because a problem is impossible doesn't mean it's difficult...

Overlaps
problem
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Overlaps

Age
11 to 14
Challenge level
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Can you find ways to put numbers in the overlaps so the rings have equal totals?
More Number Pyramids
problem
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More Number Pyramids

Age
11 to 14
Challenge level
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When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Reversals
problem
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Reversals

Age
11 to 14
Challenge level
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Where should you start, if you want to finish back where you started?
Seven Squares - Group-worthy Task
problem
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Seven Squares - Group-worthy Task

Age
11 to 14
Challenge level
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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?
Pythagoras Proofs
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Pythagoras Proofs

Age
11 to 16
Challenge level
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Can you make sense of these three proofs of Pythagoras' Theorem?

Same length
problem
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Same length

Age
11 to 16
Challenge level
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Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?
Fibonacci Surprises
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Fibonacci Surprises

Age
11 to 14
Challenge level
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Play around with the Fibonacci sequence and discover some surprising results!

Tower of Hanoi
problem
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Tower of Hanoi

Age
11 to 14
Challenge level
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The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
Star Polygons
problem
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Star Polygons

Age
11 to 14
Challenge level
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Draw some stars and measure the angles at their points. Can you find and prove a result about their sum?

Differences
problem
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Differences

Age
11 to 14
Challenge level
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Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?
Consecutive negative numbers
problem
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Consecutive negative numbers

Age
11 to 14
Challenge level
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Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
Shopping Basket
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Shopping Basket

Age
11 to 16
Challenge level
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The items in the shopping basket add and multiply to give the same amount. What could their prices be?
the greedy algorithm
problem
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the greedy algorithm

Age
11 to 14
Challenge level
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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.
Common Divisor
problem

Common Divisor

Age
14 to 18
Challenge level
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Can you find out what numbers divide these expressions? Can you prove that they are always divisors?

Generating Triples
problem
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Generating Triples

Age
14 to 16
Challenge level
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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?
How old am I?
problem
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How old am I?

Age
14 to 16
Challenge level
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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?
Finding factors
problem
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Finding factors

Age
14 to 16
Challenge level
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Can you find the hidden factors which multiply together to produce each quadratic expression?
Factorising with Multilink
problem
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Factorising with Multilink

Age
14 to 16
Challenge level
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Can you find out what is special about the dimensions of rectangles you can make with squares, sticks and units?
Nutrition and Cycling
problem
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Nutrition and Cycling

Age
14 to 16
Challenge level
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Andy wants to cycle from Land's End to John o'Groats. Will he be able to eat enough to keep him going?
Double Trouble
problem
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Double Trouble

Age
14 to 16
Challenge level
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Simple additions can lead to intriguing results...
Mega Quadratic Equations
problem

Mega Quadratic Equations

Age
14 to 18
Challenge level
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What do you get when you raise a quadratic to the power of a quadratic?
Curvy areas
problem
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Curvy areas

Age
14 to 16
Challenge level
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Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Speeding boats
problem
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Speeding boats

Age
14 to 16
Challenge level
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Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?
circles in quadrilaterals
problem
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Circles in quadrilaterals

Age
14 to 16
Challenge level
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Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

Salinon
problem
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Salinon

Age
14 to 16
Challenge level
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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?
A little light thinking
problem
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A little light thinking

Age
14 to 16
Challenge level
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Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

Doesn't add up
problem
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Doesn't add up

Age
14 to 16
Challenge level
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In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?

Quad in Quad
problem
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Quad in Quad

Age
14 to 18
Challenge level
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Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?
Multiplication arithmagons
problem
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Multiplication arithmagons

Age
14 to 16
Challenge level
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Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

Why 24?
problem
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Why 24?

Age
14 to 16
Challenge level
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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.
Iff
problem
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Iff

Age
14 to 18
Challenge level
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Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?
LCM Sudoku
problem
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LCM Sudoku

Age
14 to 16
Challenge level
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Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
CD Heaven
problem
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CD Heaven

Age
14 to 16
Challenge level
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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?
Kite in a Square
problem
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Kite in a Square

Age
14 to 18
Challenge level
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Can you make sense of the three methods to work out what fraction of the total area is shaded?
Repetitiously
problem
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Repetitiously

Age
14 to 16
Challenge level
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Can you express every recurring decimal as a fraction?
Terminology
problem
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Terminology

Age
14 to 16
Challenge level
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Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
Pythagoras Perimeters
problem
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Pythagoras Perimeters

Age
14 to 16
Challenge level
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If you know the perimeter of a right angled triangle, what can you say about the area?
Always Perfect
problem
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Always Perfect

Age
14 to 18
Challenge level
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Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
Back fitter
problem
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Back fitter

Age
14 to 18
Challenge level
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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?

Difference of Two Squares
problem
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Difference of Two Squares

Age
14 to 16
Challenge level
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What is special about the difference between squares of numbers adjacent to multiples of three?
Painted Cube
problem
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Painted Cube

Age
14 to 16
Challenge level
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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?
Puzzling Place Value
problem
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Puzzling Place Value

Age
14 to 16
Challenge level
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Can you explain what is going on in these puzzling number tricks?
Partly Painted Cube
problem
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Partly Painted Cube

Age
14 to 16
Challenge level
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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?
Multiplication square
problem
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Multiplication square

Age
14 to 16
Challenge level
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Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Picture Story
problem
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Picture Story

Age
14 to 16
Challenge level
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Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
Latin Numbers
problem
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Latin Numbers

Age
14 to 16
Challenge level
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Can you create a Latin Square from multiples of a six digit number?
Harmonic Triangle
problem
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Harmonic Triangle

Age
14 to 16
Challenge level
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Can you see how to build a harmonic triangle? Can you work out the next two rows?
2-Digit Square
problem
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2-Digit Square

Age
14 to 16
Challenge level
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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?
Difference Sudoku
problem
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Difference Sudoku

Age
14 to 16
Challenge level
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Use the differences to find the solution to this Sudoku.
Angle Trisection
problem
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Angle Trisection

Age
14 to 16
Challenge level
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It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.
Parabella
problem
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Parabella

Age
16 to 18
Challenge level
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This is a beautiful result involving a parabola and parallels.

Direct logic
problem
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Direct logic

Age
16 to 18
Challenge level
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Can you work through these direct proofs, using our interactive proof sorters?
Integration matcher
problem

Integration matcher

Age
16 to 18
Challenge level
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Can you match the charts of these functions to the charts of their integrals?