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Broad Topics > Information and Communications Technology > smartphone

### Harmonic Triangle

##### Stage: 4 Challenge Level:

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

### Areas and Ratios

##### Stage: 4 Challenge Level:

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

### Keep it Simple

##### Stage: 3 Challenge Level:

Can all unit fractions be written as the sum of two unit fractions?

### Always Perfect

##### Stage: 4 Challenge Level:

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

### Orbiting Billiard Balls

##### Stage: 4 Challenge Level:

What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?

### What's Possible?

##### Stage: 4 Challenge Level:

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

### Think of Two Numbers

##### Stage: 3 Challenge Level:

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

### Plus Minus

##### Stage: 4 Challenge Level:

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

### How Many Miles to Go?

##### Stage: 3 Challenge Level:

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?

##### Stage: 4 Challenge Level:

If a sum invested gains 10% each year how long before it has doubled its value?

### Salinon

##### Stage: 4 Challenge Level:

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?

### Beelines

##### Stage: 4 Challenge Level:

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

### The Greedy Algorithm

##### Stage: 3 Challenge Level:

The Egyptians expressed all fractions as the sum of different unit fractions. The Greedy Algorithm might provide us with an efficient way of doing this.

### Pair Products

##### Stage: 4 Challenge Level:

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

### Odd Differences

##### Stage: 4 Challenge Level:

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

### Days and Dates

##### Stage: 3 Challenge Level:

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

### Areas of Parallelograms

##### Stage: 4 Challenge Level:

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

### Nicely Similar

##### Stage: 4 Challenge Level:

If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

### Matchless

##### Stage: 4 Challenge Level:

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

### How Old Am I?

##### Stage: 4 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?

### Why 24?

##### Stage: 4 Challenge Level:

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.

### Consecutive Seven

##### Stage: 3 Challenge Level:

Can you arrange these numbers into 7 subsets, each of three numbers, so that when the numbers in each are added together, they make seven consecutive numbers?

### Perfectly Square

##### Stage: 4 Challenge Level:

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

### Sitting Pretty

##### Stage: 4 Challenge Level:

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?

### Children at Large

##### Stage: 3 Challenge Level:

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

### Fair Shares?

##### Stage: 4 Challenge Level:

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?

### Fractions Jigsaw

##### Stage: 3 Challenge Level:

A jigsaw where pieces only go together if the fractions are equivalent.

### Take Three from Five

##### Stage: 4 Challenge Level:

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?

### What's it Worth?

##### Stage: 3 and 4 Challenge Level:

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

### CD Heaven

##### Stage: 4 Challenge Level:

All CD Heaven stores were given the same number of a popular CD to sell for £24. In their two week sale each store reduces the price of the CD by 25% ... How many CDs did the store sell at. . . .

### Route to Infinity

##### Stage: 3 Challenge Level:

Can you describe this route to infinity? Where will the arrows take you next?

### One and Three

##### Stage: 4 Challenge Level:

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. . . .

### 1 Step 2 Step

##### Stage: 3 Challenge Level:

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?

### Terminology

##### Stage: 4 Challenge Level:

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

### Mirror, Mirror...

##### Stage: 3 Challenge Level:

Explore the effect of reflecting in two parallel mirror lines.

### On the Edge

##### Stage: 3 Challenge Level:

If you move the tiles around, can you make squares with different coloured edges?

### Temperature

##### Stage: 3 Challenge Level:

Water freezes at 0°Celsius (32°Fahrenheit) and boils at 100°C (212°Fahrenheit). Is there a temperature at which Celsius and Fahrenheit readings are the same?

### Who Is the Fairest of Them All ?

##### Stage: 3 Challenge Level:

Explore the effect of combining enlargements.

### Eight Hidden Squares

##### Stage: 2 and 3 Challenge Level:

On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

### Marbles in a Box

##### Stage: 3 Challenge Level:

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

### Tet-trouble

##### Stage: 4 Challenge Level:

Show that is it impossible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units...

##### Stage: 3 Challenge Level:

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

### Handshakes

##### Stage: 3 Challenge Level:

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

### 2-digit Square

##### Stage: 4 Challenge Level:

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?

### Elevenses

##### Stage: 3 Challenge Level:

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

### Differences

##### Stage: 3 Challenge Level:

Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?

### Picturing Square Numbers

##### Stage: 3 Challenge Level:

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

### Sweet Shop

##### Stage: 3 Challenge Level:

Five children went into the sweet shop after school. There were choco bars, chews, mini eggs and lollypops, all costing under 50p. Suggest a way in which Nathan could spend all his money.

### An Unusual Shape

##### Stage: 3 Challenge Level:

Can you maximise the area available to a grazing goat?

### Mixing Paints

##### Stage: 3 Challenge Level:

A decorator can buy pink paint from two manufacturers. What is the least number he would need of each type in order to produce different shades of pink.