There are 82 NRICH Mathematical resources connected to Making and proving conjectures, you may find related items under Thinking Mathematically.
Broad Topics > Thinking Mathematically > Making and proving conjectures
Always, Sometimes or Never? Shape
Age 7 to 11
Challenge Level
Are these statements always true, sometimes true or never true?
Always, Sometimes or Never? Number
Age 7 to 11
Challenge Level
Are these statements always true, sometimes true or never true?
Always, Sometimes or Never? KS1
Age 5 to 7
Challenge Level
Are these statements relating to calculation and properties of shapes always true, sometimes true or never true?
Always, Sometimes or Never?
Age 5 to 11
Challenge Level
Are these statements relating to odd and even numbers always true, sometimes true or never true?
Move Those Halves
Age 7 to 11
Challenge Level
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 ...
Six Numbered Cubes
Age 7 to 11
Challenge Level
This task combines spatial awareness with addition and multiplication.
Six Ten Total
Age 7 to 11
Challenge Level
This challenge combines addition, multiplication, perseverance and even proof.
Division Rules
Age 7 to 11
Challenge Level
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
Three Neighbours
Age 7 to 11
Challenge Level
Look at three 'next door neighbours' amongst the counting numbers. Add them together. What do you notice?
Walking Round a Triangle
Age 5 to 7
Challenge Level
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?
Square Subtraction
Age 7 to 11
Challenge Level
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?
Take Three Numbers
Age 7 to 11
Challenge Level
What happens when you add three numbers together? Will your answer be odd or even? How do you know?
What Was in the Box?
Age 5 to 7
Challenge Level
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?
Exploring Quadratic Mappings
Age 14 to 16
Challenge Level
Explore the relationship between quadratic functions and their graphs.
Exploring Simple Mappings
Age 11 to 14
Challenge Level
Explore the relationship between simple linear functions and their graphs.
Three Dice
Age 7 to 11
Challenge Level
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
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?
Magic Vs
Age 7 to 11
Challenge Level
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
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?
Spaces for Exploration
Age 11 to 14
Alf Coles writes about how he tries to create 'spaces for exploration' for the students in his classrooms.
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?
Discrete Trends
Age 16 to 18
Challenge Level
Find the maximum value of n to the power 1/n and prove that it is a maximum.
Sixty-seven Squared
Age 16 to 18
Challenge Level
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?
What's Possible?
Age 14 to 16
Challenge Level
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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?
Quad in Quad
Age 14 to 16
Challenge Level
Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?
Neighbourly Addition
Age 7 to 14
Challenge Level
I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?
Sticky Dice
Age 7 to 11
Challenge Level
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?
Diagonal in a Spiral
Age 7 to 11
Challenge Level
Investigate the totals you get when adding numbers on the diagonal of this pattern in threes.
Open Squares
Age 7 to 11
Challenge Level
This task, written for the National Young Mathematicians' Award 2016, focuses on 'open squares'. What would the next five open squares look like?
Purposeful Paper Folding
Age 5 to 11
In this article for primary teachers, Fran describes her passion for paper folding as a springboard for mathematics.
Roll over the Dice
Age 7 to 11
Challenge Level
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?
Maxagon
Age 11 to 14
Challenge Level
What's the greatest number of sides a polygon on a dotty grid could have?
Regular Hexagon Loops
Age 11 to 14
Challenge Level
Make some loops out of regular hexagons. What rules can you discover?
Spirals, Spirals
Age 7 to 11
Challenge Level
Here are two kinds of spirals for you to explore. What do you notice?
Take One Example
Age 5 to 11
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.
Binomial Coefficients
Age 14 to 18
An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.
On the Importance of Pedantry
Age 16 to 18
A introduction to how patterns can be deceiving, and what is and is not a proof.
Few and Far Between?
Age 16 to 18
Challenge Level
Can you find some Pythagorean Triples where the two smaller numbers differ by 1?
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?
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...
Painting by Numbers
Age 16 to 18
Challenge Level
How many different colours of paint would be needed to paint these pictures by numbers?
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?
Close to Triangular
Age 14 to 16
Challenge Level
Drawing a triangle is not always as easy as you might think!
The Clue Is in the Question
Age 16 to 18
Challenge Level
Starting with one of the mini-challenges, how many of the other mini-challenges will you invent for yourself?
Prime Sequences
Age 16 to 18
Challenge Level
This group tasks allows you to search for arithmetic progressions in the prime numbers. How many of the challenges will you discover for yourself?
Tiling
Age 7 to 11
Challenge Level
An investigation that gives you the opportunity to make and justify predictions.
Least of All
Age 16 to 18
Challenge Level
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.
Problem Solving, Using and Applying and Functional Mathematics
Age 5 to 18
Challenge Level
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.
NRICH is part of the family of activities in the Millennium Mathematics Project.