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### Who Is the Fairest of Them All ?

##### Stage: 3 Challenge Level:

Explore the effect of combining enlargements.

### Mirror, Mirror...

##### Stage: 3 Challenge Level:

Explore the effect of reflecting in two parallel mirror lines.

### 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...

### 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.

### Keep it Simple

##### Stage: 3 Challenge Level:

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

### 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?

### Make 37

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

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

### Litov's Mean Value Theorem

##### Stage: 3 Challenge Level:

Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...

### Searching for Mean(ing)

##### Stage: 3 Challenge Level:

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?

### Route to Infinity

##### Stage: 3 Challenge Level:

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

### Painted Cube

##### Stage: 3 Challenge Level:

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?

### Marbles in a Box

##### Stage: 3 Challenge Level:

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

### 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?

### Rule of Three

##### Stage: 3 Challenge Level:

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

### Squares in Rectangles

##### Stage: 3 Challenge Level:

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?

### 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?

### Largest Product

##### Stage: 3 Challenge Level:

Which set of numbers that add to 10 have the largest product?

### 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?

### Legs Eleven

##### Stage: 3 Challenge Level:

Take any four digit number. Move the first digit to the 'back of the queue' and move the rest along. Now add your two numbers. What properties do your answers always have?

### 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?

### Weights

##### Stage: 3 Challenge Level:

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

### 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?

### Cuboid Challenge

##### Stage: 3 Challenge Level:

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

### 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?

### Consecutive Negative Numbers

##### Stage: 3 Challenge Level:

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

### 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?

### How Much Can We Spend?

##### Stage: 3 Challenge Level:

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

### An Unusual Shape

##### Stage: 3 Challenge Level:

Can you maximise the area available to a grazing goat?

### On the Edge

##### Stage: 3 Challenge Level:

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

### Consecutive Numbers

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

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

### Summing Consecutive Numbers

##### Stage: 3 Challenge Level:

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

### Sissa's Reward

##### Stage: 3 Challenge Level:

Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...

### 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?

### Counting Factors

##### Stage: 3 Challenge Level:

Is there an efficient way to work out how many factors a large number has?

### 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?

### 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?

##### 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.

### 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 ?

### Sending a Parcel

##### Stage: 3 Challenge Level:

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

##### Stage: 3 Challenge Level:

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

### Attractive Rotations

##### Stage: 3 Challenge Level:

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

### 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?

### 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?

### 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?

##### Stage: 3 Challenge Level:

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. . . .

### Dozens

##### Stage: 3 Challenge Level:

Do you know a quick way to check if a number is a multiple of two? How about three, four or six?

### Areas of Parallelograms

##### Stage: 4 Challenge Level:

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

### Curvy Areas

##### Stage: 4 Challenge Level:

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