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Temperature

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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?

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Terminology

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Triangle ABC is isosceles while triangle DEF is equilateral. Find one angle in terms of the other two.

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Mixing More Paints

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Is it always possible to combine two paints made up in the ratios 1:x and 1:y and turn them into paint made up in the ratio a:b ? Can you find an efficent way of doing this?

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Areas and Ratios

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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.

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How Many Miles to Go?

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

A car's milometer reads 4631 miles and the trip meter has 173.3 on it. How many more miles must the car travel before the two numbers contain the same digits in the same order?

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Mixing Paints

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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.

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What's Possible?

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Beelines

Stage: 4 Challenge Level: Challenge Level:1

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

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Letter Land

Stage: 3 Challenge Level: Challenge Level:1

If: A + C = A; F x D = F; B - G = G; A + H = E; B / H = G; E - G = F and A-H represent the numbers from 0 to 7 Find the values of A, B, C, D, E, F and H.

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Pair Products

Stage: 4 Challenge Level: Challenge Level:1

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

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Take Three from Five

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

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?

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Fractions Jigsaw

Stage: 3 Challenge Level: Challenge Level:1

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

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Where to Land

Stage: 4 Challenge Level: Challenge Level:1

Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?

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Eight Hidden Squares

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

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?

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Nicely Similar

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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?

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Inscribed in a Circle

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?

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Quadrilaterals Game

Stage: 3 Challenge Level: Challenge Level:1

A game for 2 or more people, based on the traditional card game Rummy. Players aim to make two `tricks', where each trick has to consist of a picture of a shape, a name that describes that shape, and. . . .

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Elevenses

Stage: 3 Challenge Level: Challenge Level:1

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

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Weights

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Different combinations of the weights available allow you to make different totals. Which totals can you make?

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Children at Large

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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

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1 Step 2 Step

Stage: 3 Challenge Level: Challenge Level:1

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?

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How Old Am I?

Stage: 4 Challenge Level: Challenge Level:1

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?

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Matchless

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

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 ?

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Picturing Square Numbers

Stage: 3 Challenge Level: Challenge Level:1

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

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Handshakes

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Sitting Pretty

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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?

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Squares in Rectangles

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

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Areas of Parallelograms

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Mirror, Mirror...

Stage: 3 Challenge Level: Challenge Level:1

Explore the effect of reflecting in two parallel mirror lines.

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Painted Cube

Stage: 3 Challenge Level: Challenge Level:1

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?

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Litov's Mean Value Theorem

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Start with two numbers. This is the start of a sequence. The next number is the average of the last two numbers. Continue the sequence. What will happen if you carry on for ever?

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Arclets

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes".

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Searching for Mean(ing)

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

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Cuboid Challenge

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

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Sending a Parcel

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?

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Do You Feel Lucky?

Stage: 3 and 4 Challenge Level: Challenge Level:1

Some people offer advice on how to win at games of chance, or how to influence probability in your favour. Can you decide whether advice is good or not?

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In a Box

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Chris and Jo put two red and four blue ribbons in a box. They each pick a ribbon from the box without looking. Jo wins if the two ribbons are the same colour. Is the game fair?

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Cola Can

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?

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Salinon

Stage: 4 Challenge Level: Challenge Level:1

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?

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M, M and M

Stage: 3 Challenge Level: Challenge Level:1

If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?

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Route to Infinity

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Who Is the Fairest of Them All?

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Explore the effect of combining enlargements.

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Shady Symmetry

Stage: 3 Challenge Level: Challenge Level:1

How many different symmetrical shapes can you make by shading triangles or squares?

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Far Horizon

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?

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Attractive Rotations

Stage: 3 Challenge Level: Challenge Level:1

Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...

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On the Edge

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Here are four tiles. They can be arranged in a 2 by 2 square so that this large square has a green edge. If the tiles are moved around, we can make a 2 by 2 square with a blue edge... Now try to. . . .

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Curvy Areas

Stage: 4 Challenge Level: Challenge Level:1

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

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An Unusual Shape

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Can you maximise the area available to a grazing goat?

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Can They Be Equal?

Stage: 3 Challenge Level: Challenge Level:1

Can you find rectangles where the value of the area is the same as the value of the perimeter?

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Orbiting Billiard Balls

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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