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There are **176** NRICH Mathematical resources connected to **Creating and manipulating expressions and formulae**, you may find related items under Algebraic expressions, equations and formulae.

Problem
Primary curriculum
Secondary curriculum
### Temperature

Is there a temperature at which Celsius and Fahrenheit readings are the same?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Sums of Pairs

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Areas of Parallelograms

Can you find the area of a parallelogram defined by two vectors?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Harmonic Triangle

Can you see how to build a harmonic triangle? Can you work out the next two rows?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cubes Within Cubes Revisited

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Partitioning Revisited

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Multiplication Square

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Diagonal Sums

In this 100 square, look at the green square which contains the numbers 2, 3, 12 and 13. What is the sum of the numbers that are diagonally opposite each other? What do you notice?

Age 7 to 14

Challenge Level

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
### Complex Partial Fractions

To break down an algebraic fraction into partial fractions in which all the denominators are linear and all the numerators are constants you sometimes need complex numbers.

Age 16 to 18

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
### Your Number Is...

Think of a number and follow the machine's instructions... I know what your number is! Can you explain how I know?

Age 7 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
### Pair Products

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Special Numbers

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

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
### How Many Miles to Go?

How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Mind Reading

Think of a number, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I know?

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
### Sums of Squares

Can you prove that twice the sum of two squares always gives the sum of two squares?

Age 16 to 18

Challenge Level

Problem
Live Primary curriculum
Secondary curriculum
### Pick's Theorem

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Terminology

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

Age 14 to 16

Challenge Level

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
### Think of Two Numbers

Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?

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
### Fair Shares?

A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Sitting Pretty

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

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
### Plus Minus

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### One and Three

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400 metres from B. How long is the lake?

Age 14 to 16

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
### System Speak

Five equations... five unknowns... can you solve the system?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Odd Differences

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = nÂ² Use the diagram to show that any odd number is the difference of two squares.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Fibs

The well known Fibonacci sequence is 1 ,1, 2, 3, 5, 8, 13, 21.... How many Fibonacci sequences can you find containing the number 196 as one of the terms?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Always Two

Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2.

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### 2-digit Square

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

Age 14 to 16

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
### 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