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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?

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?

Product Sudoku

Stage: 3 Challenge Level:

The clues for this Sudoku are the product of the numbers in adjacent squares.

Funny Factorisation

Stage: 3 Challenge Level:

Some 4 digit numbers can be written as the product of a 3 digit number and a 2 digit number using the digits 1 to 9 each once and only once. The number 4396 can be written as just such a product. Can. . . .

American Billions

Stage: 3 Challenge Level:

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

Going Round in Circles

Stage: 3 Challenge Level:

Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?

Dozens

Stage: 2 and 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?

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?

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.

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?

Counting Factors

Stage: 3 Challenge Level:

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

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?

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...

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?

Ben's Game

Stage: 3 Challenge Level:

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

Cuboids

Stage: 3 Challenge Level:

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

Expenses

Stage: 4 Challenge Level:

What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?

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.

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?

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?

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?

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...

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?

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?

Special Numbers

Stage: 3 Challenge Level:

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?

An Unusual Shape

Stage: 3 Challenge Level:

Can you maximise the area available to a grazing goat?

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?

M, M and M

Stage: 3 Challenge Level:

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

Mirror, Mirror...

Stage: 3 Challenge Level:

Explore the effect of reflecting in two parallel mirror lines.

Route to Infinity

Stage: 3 Challenge Level:

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

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?

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?

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?

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?

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?

Marbles in a Box

Stage: 3 Challenge Level:

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

On the Edge

Stage: 3 Challenge Level:

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

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?

Mixing More Paints

Stage: 3 Challenge Level:

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?

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.

Negative Power

Stage: 4 Challenge Level:

What does this number mean ? Which order of 1, 2, 3 and 4 makes the highest value ? Which makes the lowest ?

Keep it Simple

Stage: 3 Challenge Level:

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

Repetitiously

Stage: 3 Challenge Level:

The number 2.525252525252.... can be written as a fraction. What is the sum of the denominator and numerator?

Number Daisy

Stage: 3 Challenge Level:

Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?

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.

Two and Two

Stage: 3 Challenge Level:

How many solutions can you find to this sum? Each of the different letters stands for a different number.

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?

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?

Largest Product

Stage: 3 Challenge Level:

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