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There are **229** NRICH Mathematical resources connected to **Generalising**, you may find related items under Thinking mathematically.

Problem
Primary curriculum
Secondary curriculum
### More Magic Potting Sheds

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?

Age 11 to 16

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Primary curriculum
Secondary curriculum
### Areas of Parallelograms

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

Age 14 to 16

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Primary curriculum
Secondary curriculum
### Litov's Mean Value Theorem

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

Age 11 to 14

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Problem
Primary curriculum
Secondary curriculum
### Squares in Rectangles

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?

Age 11 to 14

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Primary curriculum
Secondary curriculum
### Harmonic Triangle

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

Age 14 to 16

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Problem
Primary curriculum
Secondary curriculum
### Cubes Within Cubes Revisited

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?

Age 11 to 14

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Problem
Primary curriculum
Secondary curriculum
### Partitioning Revisited

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

Age 11 to 14

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Problem
Primary curriculum
Secondary curriculum
### Multiplication Square

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

Age 14 to 16

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Primary curriculum
Secondary curriculum
### Number Differences

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

Age 7 to 11

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Problem
Primary curriculum
Secondary curriculum
### More Numbers in the Ring

If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice?

Age 5 to 7

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Problem
Primary curriculum
Secondary curriculum
### Arithmagons

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

Age 11 to 16

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Problem
Primary curriculum
Secondary curriculum
### Lots of Lollies

Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?

Age 5 to 7

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Primary curriculum
Secondary curriculum
### Painted Cube

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Tilted Squares

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

Age 11 to 14

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Problem
Primary curriculum
Secondary curriculum
### Coordinate Patterns

Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?

Age 11 to 14

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Problem
Primary curriculum
Secondary curriculum
### Seven Squares - Group-worthy Task

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?

Age 11 to 14

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Problem
Primary curriculum
Secondary curriculum
### Shear Magic

Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?

Age 11 to 14

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Problem
Primary curriculum
Secondary curriculum
### Break it Up!

In how many different ways can you break up a stick of seven interlocking cubes? Now try with a stick of eight cubes and a stick of six cubes. What do you notice?

Age 5 to 11

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Problem
Primary curriculum
Secondary curriculum
### Up and Down Staircases

One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?

Age 7 to 11

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Problem
Primary curriculum
Secondary curriculum
### More Number Pyramids

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

Age 11 to 14

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Problem
Primary curriculum
Secondary curriculum
### Number Pyramids

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

Age 11 to 14

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Problem
Primary curriculum
Secondary curriculum
### Odd Squares

Think of a number, square it and subtract your starting number. Is the number you're left with odd or even? How do the images help to explain this?

Age 7 to 11

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Problem
Primary curriculum
Secondary curriculum
### Pair Products

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

Age 14 to 16

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Problem
Primary curriculum
Secondary curriculum
### Picturing Square Numbers

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

Age 11 to 14

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Primary curriculum
Secondary curriculum
### Picturing Triangular Numbers

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

Age 11 to 14

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Problem
Primary curriculum
Secondary curriculum
### Mind Reading

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 know?

Age 11 to 14

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Problem
Primary curriculum
Secondary curriculum
### Make 37

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

Age 7 to 11

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Problem
Primary curriculum
Secondary curriculum
### Pick's Theorem

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.

Age 14 to 16

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Problem
Primary curriculum
Secondary curriculum
### Take Three from Five

Caroline and James pick sets of five numbers. Charlie tries to find three that add together to make a multiple of three. Can they stop him?

Age 11 to 16

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Problem
Primary curriculum
Secondary curriculum
### Got It

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.

Age 7 to 14

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Problem
Primary curriculum
Secondary curriculum
### Frogs

How many moves does it take to swap over some red and blue frogs? Do you have a method?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Nim-7

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

Age 5 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Egyptian Fractions

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.

Age 11 to 14

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Problem
Primary curriculum
Secondary curriculum
### Attractive Tablecloths

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?

Age 14 to 16

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Problem
Primary curriculum
Secondary curriculum
### Enclosing Squares

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

Age 11 to 14

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Primary curriculum
Secondary curriculum
### Where Can We Visit?

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?

Age 11 to 14

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Primary curriculum
Secondary curriculum
### What's Possible?

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

Age 14 to 16

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Problem
Primary curriculum
Secondary curriculum
### Beelines

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

Age 14 to 16

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Problem
Primary curriculum
Secondary curriculum
### Cut it Out

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

Age 7 to 11

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Primary curriculum
Secondary curriculum
### Of All the Areas

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

Age 14 to 16

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Problem
Primary curriculum
Secondary curriculum
### Plus Minus

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

Age 14 to 16

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Problem
Primary curriculum
Secondary curriculum
### Odd Differences

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = nÂ² Use the diagram to show that any odd number is the difference of two squares.

Age 14 to 16

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Problem
Primary curriculum
Secondary curriculum
### Summing Consecutive Numbers

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

Age 11 to 14

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Problem
Primary curriculum
Secondary curriculum
### Is There a Theorem?

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

Age 11 to 14

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Problem
Primary curriculum
Secondary curriculum
### Counting Factors

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

Age 11 to 14

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Primary curriculum
Secondary curriculum
### Have You Got It?

Can you explain the strategy for winning this game with any target?

Age 11 to 14

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Problem
Primary curriculum
Secondary curriculum
### Shape and Territory

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

Age 16 to 18

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Primary curriculum
Secondary curriculum
### Sticky Triangles

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

Age 7 to 11

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Primary curriculum
Secondary curriculum
### Round and Round the Circle

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

Age 7 to 11

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Problem
Primary curriculum
Secondary curriculum
### Doplication

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

Age 7 to 11

Challenge Level