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Broad Topics > Using, Applying and Reasoning about Mathematics > Generalising

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Partly Painted Cube

Age 14 to 16 Challenge Level:

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

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Handshakes

Age 11 to 14 Challenge Level:

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

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Mystic Rose

Age 14 to 16 Challenge Level:

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

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Tower of Hanoi

Age 11 to 14 Challenge Level:

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

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Christmas Chocolates

Age 11 to 14 Challenge Level:

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

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How Much Can We Spend?

Age 11 to 14 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?

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Keep it Simple

Age 11 to 14 Challenge Level:

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

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For Richer for Poorer

Age 14 to 16 Challenge Level:

Charlie has moved between countries and the average income of both has increased. How can this be so?

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Cuboid Challenge

Age 11 to 14 Challenge Level:

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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Elevenses

Age 11 to 14 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?

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Searching for Mean(ing)

Age 11 to 14 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?

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Consecutive Negative Numbers

Age 11 to 14 Challenge Level:

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

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More Twisting and Turning

Age 11 to 14 Challenge Level:

It would be nice to have a strategy for disentangling any tangled ropes...

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Sums of Pairs

Age 11 to 16 Challenge Level:

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

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Triangle Numbers

Age 11 to 14 Challenge Level:

Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?

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Route to Infinity

Age 11 to 14 Challenge Level:

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

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Cunning Card Trick

Age 11 to 14 Challenge Level:

Delight your friends with this cunning trick! Can you explain how it works?

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Who Is the Fairest of Them All ?

Age 11 to 14 Challenge Level:

Explore the effect of combining enlargements.

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...on the Wall

Age 11 to 14 Challenge Level:

Explore the effect of reflecting in two intersecting mirror lines.

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Mirror, Mirror...

Age 11 to 14 Challenge Level:

Explore the effect of reflecting in two parallel mirror lines.

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More Magic Potting Sheds

Age 11 to 14 Challenge Level:

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

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Areas of Parallelograms

Age 14 to 16 Challenge Level:

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

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Litov's Mean Value Theorem

Age 11 to 14 Challenge Level:

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

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Squares in Rectangles

Age 11 to 14 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?

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Harmonic Triangle

Age 14 to 16 Challenge Level:

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

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Cubes Within Cubes Revisited

Age 11 to 14 Challenge Level:

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

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Partitioning Revisited

Age 11 to 14 Challenge Level:

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

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Multiplication Square

Age 14 to 16 Challenge Level:

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

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Arithmagons

Age 14 to 16 Challenge Level:

Can you find the values at the vertices when you know the values on the edges?

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Painted Cube

Age 11 to 14 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?

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Tilted Squares

Age 11 to 14 Challenge Level:

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

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Seven Squares - Group-worthy Task

Age 11 to 14 Challenge Level:

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

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More Number Pyramids

Age 11 to 14 Challenge Level:

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

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Number Pyramids

Age 11 to 14 Challenge Level:

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

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Pair Products

Age 14 to 16 Challenge Level:

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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Picturing Square Numbers

Age 11 to 14 Challenge Level:

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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Picturing Triangular Numbers

Age 11 to 14 Challenge Level:

Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

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Mind Reading

Age 11 to 14 Challenge Level:

Think of a number, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I. . . .

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Make 37

Age 7 to 14 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.

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Pick's Theorem

Age 11 to 14 Challenge Level:

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

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Take Three from Five

Age 14 to 16 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?

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Got It

Age 7 to 14 Challenge Level:

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

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Nim-7

Age 5 to 14 Challenge Level:

Can you work out how to win this game of Nim? Does it matter if you go first or second?

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Egyptian Fractions

Age 11 to 14 Challenge Level:

The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.

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Attractive Tablecloths

Age 14 to 16 Challenge Level:

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

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Enclosing Squares

Age 11 to 14 Challenge Level:

Can you find sets of sloping lines that enclose a square?

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Where Can We Visit?

Age 11 to 14 Challenge Level:

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

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What's Possible?

Age 14 to 16 Challenge Level:

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

Age 14 to 16 Challenge Level:

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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Of All the Areas

Age 14 to 16 Challenge Level:

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?