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Broad Topics > Using, Applying and Reasoning about Mathematics > Generalising Route to Infinity

Age 11 to 14 Challenge Level:

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

Age 11 to 14 Challenge Level:

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board? Hidden Rectangles

Age 11 to 14 Challenge Level:

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard? 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? Tourism

Age 11 to 14 Challenge Level:

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. Window Frames

Age 5 to 14 Challenge Level:

This task encourages you to investigate the number of edging pieces and panes in different sized windows. 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? Dotty Triangles

Age 11 to 14 Challenge Level:

Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw? 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? 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? 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? Konigsberg Plus

Age 11 to 14 Challenge Level:

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. Cunning Card Trick

Age 11 to 14 Challenge Level:

Delight your friends with this cunning trick! Can you explain how it works? 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? Card Trick 2

Age 11 to 14 Challenge Level:

Can you explain how this card trick works? Is There a Theorem?

Age 11 to 14 Challenge Level:

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? 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? Frogs

Age 11 to 14 Challenge Level:

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

Age 14 to 16 Challenge Level:

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? More Number Pyramids

Age 11 to 14 Challenge Level:

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

Age 11 to 14 Challenge Level:

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13. Age 11 to 14 Challenge Level:

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? 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? Jam

Age 14 to 16 Challenge Level:

A game for 2 players Squares, Squares and More Squares

Age 11 to 14 Challenge Level:

Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares? 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. . . . Three Times Seven

Age 11 to 14 Challenge Level:

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why? ...on the Wall

Age 11 to 14 Challenge Level:

Explore the effect of reflecting in two intersecting mirror lines. Enclosing Squares

Age 11 to 14 Challenge Level:

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

Age 7 to 16

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

Age 11 to 14

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important. Mini-max

Age 11 to 14 Challenge Level:

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. . . . Sum Equals Product

Age 11 to 14 Challenge Level:

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. . . . Special Sums and Products

Age 11 to 14 Challenge Level:

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. Regular Hexagon Loops

Age 11 to 14 Challenge Level:

Make some loops out of regular hexagons. What rules can you discover? Keep it Simple

Age 11 to 14 Challenge Level:

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

Age 14 to 16 Challenge Level:

To avoid losing think of another very well known game where the patterns of play are similar. Who Is the Fairest of Them All ?

Age 11 to 14 Challenge Level:

Explore the effect of combining enlargements. Nim-7 for Two

Age 5 to 14 Challenge Level:

Nim-7 game for an adult and child. Who will be the one to take the last counter? Age 11 to 14 Challenge Level:

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. . . . Steps to the Podium

Age 7 to 14 Challenge Level:

It starts quite simple but great opportunities for number discoveries and patterns! Magic Letters

Age 11 to 14 Challenge Level:

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?