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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?

Think of Two Numbers

Stage: 3 Challenge Level:

Think of two whole numbers under 10. Take one of them and add 1. Multiply by 5. Add 1 again. Double your answer. Subract 1. Add your second number. Add 2. Double again. Subtract 8. Halve this number. . . .

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?

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.

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?

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?

Plus Minus

Stage: 4 Challenge Level:

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

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.

Terminology

Stage: 3 Challenge Level:

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

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?

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?

How Many Miles to Go?

Stage: 3 Challenge Level:

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?

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.

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?

Keep it Simple

Stage: 3 Challenge Level:

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

Going Round in Circles

Stage: 3 Challenge Level:

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

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?

Who Is the Fairest of Them All?

Stage: 3 Challenge Level:

Explore the effect of combining enlargements.

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. . . .

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?

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?

Take Three from Five

Stage: 3 and 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?

Mirror, Mirror...

Stage: 3 Challenge Level:

Explore the effect of reflecting in two parallel mirror lines.

14 Divisors

Stage: 3 Challenge Level:

What is the smallest number with exactly 14 divisors?

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?

Consecutive Negative Numbers

Stage: 3 Challenge Level:

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

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?

Litov's Mean Value Theorem

Stage: 3 Challenge Level:

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?

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?

Areas of Parallelograms

Stage: 4 Challenge Level:

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

Route to Infinity

Stage: 3 and 4 Challenge Level:

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

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.

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?

Marbles in a Box

Stage: 3 and 4 Challenge Level:

In a three-dimensional version of noughts and crosses, how many winning lines can you make?

Matchless

Stage: 3 and 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 ?

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?

Weights

Stage: 3 Challenge Level:

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

Days and Dates

Stage: 4 Challenge Level:

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

Perfectly Square

Stage: 4 Challenge Level:

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

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?

Inscribed in a Circle

Stage: 3 Challenge Level:

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?

An Unusual Shape

Stage: 3 Challenge Level:

Can you maximise the area available to a grazing goat?

On the Edge

Stage: 3 Challenge Level:

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. . . .

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?

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. . . .

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?

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?

Repetitiously

Stage: 3 Challenge Level:

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

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?