There are **255** NRICH Mathematical resources connected to **Mathematical reasoning & proof**, you may find related items under Thinking Mathematically.

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

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

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

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

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

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?

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?

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

This problem looks at how one example of your choice can show something about the general structure of multiplication.

This investigates one particular property of number by looking closely at an example of adding two odd numbers together.

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

Can you make sense of these three proofs of Pythagoras' Theorem?

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

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

Can you work through these direct proofs, using our interactive proof sorters?

Can you rearrange the cards to make a series of correct mathematical statements?

Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number?

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

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?

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?

The sums of the squares of three related numbers is also a perfect square - can you explain why?

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?

A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.

Which set of numbers that add to 10 have the largest product?

Put the steps of this proof in order to find the formula for the sum of an arithmetic sequence

This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.

Some diagrammatic 'proofs' of algebraic identities and inequalities.

Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.

Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?

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.

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line?

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.

To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.

What can you say about the common difference of an AP where every term is prime?

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.