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Resources tagged with Generalising similar to Euler's Formula and Topology:

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Magic Squares II

Stage: 4 and 5

An article which gives an account of some properties of magic squares.

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AMGM

Stage: 4 Challenge Level: Challenge Level:1

Choose any two numbers. Call them a and b. Work out the arithmetic mean and the geometric mean. Which is bigger? Repeat for other pairs of numbers. What do you notice?

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Tourism

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

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Janine's Conjecture

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

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How Much Can We Spend?

Stage: 3 Challenge Level: Challenge Level:1

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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Konigsberg Plus

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

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Multiplication Square

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Problem Solving, Using and Applying and Functional Mathematics

Stage: 1, 2, 3, 4 and 5 Challenge Level: Challenge Level:1

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

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Adding in Rows

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

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Jam

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

This is a Dutch game for two players. It will test your powers of shape and space visualisation

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Where Can We Visit?

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Charlie and Lynne 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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Sums of Pairs

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

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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More Number Pyramids

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Are there any patterns within the pyramid? Can you explain why you only get multiples of 4 at the top when you start with an integer in the bottom left hand corner?

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Cubes Within Cubes Revisited

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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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Go Forth and Generalise

Stage: 3

This article begins to look at what it means to generalise and the importance of looking beyond spotting patterns to understanding why the patterns are there.

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Jam - a Game for Two Players

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

A game for 2 players

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Number Pyramids

Stage: 3 Challenge Level: Challenge Level:1

Using the same starter numbers 2, 1, 4 and 6 can you get a larger total at the top of the pyramid? What is the largest total you can get?

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Chocolate 2004

Stage: 4 Challenge Level: Challenge Level:1

First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...

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Pentanim,a Game for Two Players

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

A game for 2 players with similaritlies to NIM. Place one counter on each spot on the games board. Players take it is turns to remove 1 or 2 adjacent counters. The winner picks up the last counter.

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Nim-interactive

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The winner is the player to take the last counter.

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Partitioning Revisited

Stage: 3 Challenge Level: Challenge Level:1

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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Areas of Parallelograms

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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Pair Products

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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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Games Related to Nim

Stage: 1, 2, 3 and 4

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

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Mindreader

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .

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Winning Lines

Stage: 2, 3 and 4 Challenge Level: Challenge Level:1

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.

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Sliding Puzzle

Stage: 1, 2, 3 and 4 Challenge Level: Challenge Level:1

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

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Sum Equals Product

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .

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Christmas Chocolates

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Lower Bound

Stage: 3 Challenge Level: Challenge Level:1

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

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One, Three, Five, Seven

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.

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Magic Squares

Stage: 4 and 5

An account of some magic squares and their properties and and how to construct them for yourself.

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Building Gnomons

Stage: 4 Challenge Level: Challenge Level:1

Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.

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Lattice Points

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?

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Chocolate Maths

Stage: 3 Challenge Level: Challenge Level:1

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

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Mini-max

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers. . . .

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Enclosing Squares

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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Converging Means

Stage: 3 Challenge Level: Challenge Level:1

Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .

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Picturing Square Numbers

Stage: 3 Challenge Level: Challenge Level:1

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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Painted Cube

Stage: 3 Challenge Level: Challenge Level:1

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. Can you predict how many of the faces of the smaller cubes will remain red?

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Number Tricks

Stage: 3 Challenge Level: Challenge Level:1

Think of a number add 3 double add 4 halve take away the number you started with ? What did you end up with? Now try again starting with a different number. Try again? Try starting with a fraction. . . .

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Picturing Triangle Numbers

Stage: 3 Challenge Level: Challenge Level:1

Triangle 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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Special Sums and Products

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

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Frogs

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

You can solve frogs on the computer, using counters, or acting it out. Start with frogs in a line on one side, and toads on the other, with a space in between. They need to change places.

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Take Three from Five

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Can you guarantee that from any group of 5 numbers it is always possible to choose three numbers that will add up to a multiple of 3?

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Keep it Simple

Stage: 3 Challenge Level: Challenge Level:1

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

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Cuboid Challenge

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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

Stage: 3 Challenge Level: Challenge Level:1

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)

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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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Egyptian Fractions

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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.