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Broad Topics > Using, Applying and Reasoning about Mathematics > Making and proving conjectures

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What Was in the Box?

Stage: 1 Challenge Level: Challenge Level:1

This big box adds something to any number that goes into it. If you know the numbers that come out, what addition might be going on in the box?

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Exploring Quadratic Mappings

Stage: 4 Challenge Level: Challenge Level:1

Explore the relationship between quadratic functions and their graphs.

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Exploring Simple Mappings

Stage: 3 Challenge Level: Challenge Level:1

Explore the relationship between simple linear functions and their graphs.

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Curvy Areas

Stage: 4 Challenge Level: Challenge Level:1

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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Magic Vs

Stage: 2 Challenge Level: Challenge Level:2 Challenge Level:2

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

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Consecutive Negative Numbers

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

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Epidemic Modelling

Stage: 4 and 5 Challenge Level: Challenge Level:1

Use the computer to model an epidemic. Try out public health policies to control the spread of the epidemic, to minimise the number of sick days and deaths.

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Multiplication Square

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

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Discrete Trends

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Find the maximum value of n to the power 1/n and prove that it is a maximum.

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Sixty-seven Squared

Stage: 5 Challenge Level: Challenge Level:1

Evaluate these powers of 67. What do you notice? Can you convince someone what the answer would be to (a million sixes followed by a 7) squared?

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What's Possible?

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

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How Old Am I?

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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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Pericut

Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2

Two semicircle sit on the diameter of a semicircle centre O of twice their radius. Lines through O divide the perimeter into two parts. What can you say about the lengths of these two parts?

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Division Rules

Stage: 2 Challenge Level: Challenge Level:1

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

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Three Neighbours

Stage: 2 Challenge Level: Challenge Level:2 Challenge Level:2

Look at three 'next door neighbours' amongst the counting numbers. Add them together. What do you notice?

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Walking Round a Triangle

Stage: 1 Challenge Level: Challenge Level:1

This ladybird is taking a walk round a triangle. Can you see how much he has turned when he gets back to where he started?

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Take One Example

Stage: 1 and 2

This article introduces the idea of generic proof for younger children and illustrates how one example can offer a proof of a general result through unpacking its underlying structure.

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Square Subtraction

Stage: 2 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Look at what happens when you take a number, square it and subtract your answer. What kind of number do you get? Can you prove it?

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Take Three Numbers

Stage: 2 Challenge Level: Challenge Level:1

What happens when you add three numbers together? Will your answer be odd or even? How do you know?

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On the Importance of Pedantry

Stage: 3, 4 and 5

A introduction to how patterns can be deceiving, and what is and is not a proof.

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Few and Far Between?

Stage: 4 and 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Can you find some Pythagorean Triples where the two smaller numbers differ by 1?

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Multiplication Arithmagons

Stage: 4 Challenge Level: Challenge Level:1

Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

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Always a Multiple?

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...

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Painting by Numbers

Stage: 5 Challenge Level: Challenge Level:1

How many different colours of paint would be needed to paint these pictures by numbers?

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A Little Light Thinking

Stage: 4 Challenge Level: Challenge Level:1

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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Becky's Number Plumber

Stage: 2 Challenge Level: Challenge Level:1

Becky created a number plumber which multiplies by 5 and subtracts 4. What do you notice about the numbers that it produces? Can you explain your findings?

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Steve's Mapping

Stage: 5 Challenge Level: Challenge Level:1

Steve has created two mappings. Can you figure out what they do? What questions do they prompt you to ask?

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Alison's Mapping

Stage: 4 Challenge Level: Challenge Level:1

Alison has created two mappings. Can you figure out what they do? What questions do they prompt you to ask?

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Become Maths Detectives

Stage: 2 Challenge Level: Challenge Level:1

Explore Alex's number plumber. What questions would you like to ask? Don't forget to keep visiting NRICH projects site for the latest developments and questions.

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Charlie's Mapping

Stage: 3 Challenge Level: Challenge Level:1

Charlie has created a mapping. Can you figure out what it does? What questions does it prompt you to ask?

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Close to Triangular

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Drawing a triangle is not always as easy as you might think!

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Three Dice

Stage: 2 Challenge Level: Challenge Level:1

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

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The Clue Is in the Question

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

This problem is a sequence of linked mini-challenges leading up to the proof of a difficult final challenge, encouraging you to think mathematically. Starting with one of the mini-challenges, how. . . .

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Prime Sequences

Stage: 5 Challenge Level: Challenge Level:1

This group tasks allows you to search for arithmetic progressions in the prime numbers. How many of the challenges will you discover for yourself?

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Tiling

Stage: 2 Challenge Level: Challenge Level:1

An investigation that gives you the opportunity to make and justify predictions.

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Least of All

Stage: 5 Challenge Level: Challenge Level:1

A point moves on a line segment. A function depends on the position of the point. Where do you expect the point to be for a minimum of this function to occur.

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Problem Solving, Using and Applying and Functional Mathematics

Stage: 1, 2, 3, 4 and 5 Challenge Level: Challenge Level:1

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

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Masterclasses Information

Stage: 1 and 2 Challenge Level: Challenge Level:1

If you'd like to know more about Primary Maths Masterclasses, this is the package to read! Find out about current groups in your region or how to set up your own.

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Conjugate Tracker

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Make a conjecture about the curved track taken by the complex roots of a quadratic equation and use complex conjugates to prove your conjecture.

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Integral Sandwich

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Generalise this inequality involving integrals.

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Tables Without Tens

Stage: 2 Challenge Level: Challenge Level:1

Investigate and explain the patterns that you see from recording just the units digits of numbers in the times tables.

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Vecten

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Join in this ongoing research. Build squares on the sides of a triangle, join the outer vertices forming hexagons, build further rings of squares and quadrilaterals, investigate.

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Trig Rules OK

Stage: 5 Challenge Level: Challenge Level:1

Change the squares in this diagram and spot the property that stays the same for the triangles. Explain...

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Primary Proof?

Stage: 1

Proof does have a place in Primary mathematics classrooms, we just need to be clear about what we mean by proof at this level.

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Plus or Minus

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.

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Pythagorean Fibs

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

What have Fibonacci numbers got to do with Pythagorean triples?

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Fibonacci Fashion

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

What have Fibonacci numbers to do with solutions of the quadratic equation x^2 - x - 1 = 0 ?

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Shuffles

Stage: 5 Challenge Level: Challenge Level:1

An environment for exploring the properties of small groups.

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Triangles Within Pentagons

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Show that all pentagonal numbers are one third of a triangular number.

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Triangles Within Squares

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Can you find a rule which relates triangular numbers to square numbers?