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There are **356** NRICH Mathematical resources connected to **Reasoning, justifying and proof**, you may find related items under Thinking mathematically.

Problem
Primary curriculum
Secondary curriculum
### Salinon

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Painted Cube

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Tilted Squares

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Seven Squares - Group-worthy Task

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Perfectly Square

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### More Number Pyramids

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Number Pyramids

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Always Perfect

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

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Children at Large

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Leonardo's Problem

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?

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Sixty-seven Squared

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?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Eyes Down

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?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Take Three from Five

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?

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Power Quady

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Largest Product

Which set of numbers that add to 100 have the largest product?

Age 11 to 14

Challenge Level

Interactive
Primary curriculum
Secondary curriculum
### Proof Sorter - Sum of an Arithmetic Sequence

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

Age 16 to 18

Challenge Level

Interactive
Primary curriculum
Secondary curriculum
### Proof Sorter - Quadratic Equation

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.

Age 14 to 18

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Proofs with Pictures

Some diagrammatic 'proofs' of algebraic identities and inequalities.

Age 14 to 18

Article
Primary curriculum
Secondary curriculum
### Telescoping Functions

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.

Age 16 to 18

Problem
Primary curriculum
Secondary curriculum
### What's it Worth?

There are lots of different methods to find out what the shapes are worth - how many can you find?

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### 1 Step 2 Step

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Attractive Tablecloths

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Marbles in a Box

How many winning lines can you make in a three-dimensional version of noughts and crosses?

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Number Rules - OK

Can you produce convincing arguments that a selection of statements about numbers are true?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Lens Angle

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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Iff

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

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Why 24?

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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### What's Possible?

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Angle Trisection

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Zig Zag

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Unit Interval

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

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### How Old Am I?

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Quad in Quad

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

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Legs Eleven

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Rule of Three

If it takes four men one day to build a wall, how long does it take 60,000 men to build a similar wall?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Summing Consecutive Numbers

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Quadratic Harmony

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Mechanical Integration

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Latin Numbers

Can you create a Latin Square from multiples of a six digit number?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Prime AP

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Common Divisor

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### There's a Limit

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?

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Square Mean

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cosines Rule

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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Binomial

By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Polynomial Relations

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Picture Story

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Shape and Territory

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Areas and Ratios

Do you have enough information to work out the area of the shaded quadrilateral?

Age 16 to 18

Challenge Level