There are 353 NRICH Mathematical resources connected to Reasoning, justifying and proof, you may find related items under Thinking mathematically.
Broad Topics > Thinking mathematically > Reasoning, justifying and proof
Can you find the values at the vertices when you know the values on the edges?
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?
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?
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
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...
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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?
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?
The symbol [ ] means 'the integer part of'. Can the numbers [2x]; 2[x]; [x + 1/2] + [x - 1/2] ever be equal? Can they ever take three different values?
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?
Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.
Which set of numbers that add to 100 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.
There are lots of different methods to find out what the shapes are worth - how many can you find?
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?
Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
Can you produce convincing arguments that a selection of statements about numbers are true?
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.
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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?
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?
Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?
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?
15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?
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.
Can you create a Latin Square from multiples of a six digit number?
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.
By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn
Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?
Do you have enough information to work out the area of the shaded quadrilateral?