There are **82** NRICH Mathematical resources connected to **Making and proving conjectures**, you may find related items under Thinking Mathematically.

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Are these statements always true, sometimes true or never true?

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Are these statements always true, sometimes true or never true?

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Are these statements relating to calculation and properties of shapes always true, sometimes true or never true?

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Are these statements relating to odd and even numbers always true, sometimes true or never true?

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For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...

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This task combines spatial awareness with addition and multiplication.

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This challenge combines addition, multiplication, perseverance and even proof.

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

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

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

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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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Explore the relationship between quadratic functions and their graphs.

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Explore the relationship between simple linear functions and their graphs.

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

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

Alf Coles writes about how he tries to create 'spaces for exploration' for the students in his classrooms.

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

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

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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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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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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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Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?

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I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

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Throughout these challenges, the touching faces of any adjacent dice must have the same number. Can you find a way of making the total on the top come to each number from 11 to 18 inclusive?

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Investigate the totals you get when adding numbers on the diagonal of this pattern in threes.

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This task, written for the National Young Mathematicians' Award 2016, focuses on 'open squares'. What would the next five open squares look like?

In this article for primary teachers, Fran describes her passion for paper folding as a springboard for mathematics.

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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?

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What's the greatest number of sides a polygon on a dotty grid could have?

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Make some loops out of regular hexagons. What rules can you discover?

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Here are two kinds of spirals for you to explore. What do you notice?

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.

An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.

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

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

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

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

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

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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?

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

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Starting with one of the mini-challenges, how many of the other mini-challenges will you invent for yourself?

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

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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 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.