List

Visualising and Representing - Secondary Students

The Number Jumbler
problem
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The Number Jumbler

Age
7 to 14
Challenge level
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The Number Jumbler can always work out your chosen symbol. Can you work out how?
Treasure Hunt
problem
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Treasure Hunt

Age
7 to 14
Challenge level
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Can you find a reliable strategy for choosing coordinates that will locate the treasure in the minimum number of guesses?
Subtraction Surprise
problem
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Subtraction Surprise

Age
7 to 14
Challenge level
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Try out some calculations. Are you surprised by the results?
Your number is...
problem
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Your number is...

Age
7 to 14
Challenge level
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Think of a number and follow the machine's instructions... I know what your number is! Can you explain how I know?
Method in multiplying madness?
problem
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Method in multiplying madness?

Age
7 to 14
Challenge level
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Watch our videos of multiplication methods that you may not have met before. Can you make sense of them?
Remainders
problem
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Remainders

Age
7 to 14
Challenge level
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I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?
Number Lines in Disguise
problem
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Number Lines in Disguise

Age
7 to 14
Challenge level
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Some of the numbers have fallen off Becky's number line. Can you figure out what they were?
Starting Fibonacci
problem

Starting Fibonacci

Age
11 to 14
Challenge level
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What is the first term of a Fibonacci sequence whose second term is 4 and fifth term is 22?
Parallelogram It
problem
Favourite

Parallelogram It

Age
11 to 16
Challenge level
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Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a parallelogram.

Your number was...
problem
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Your number was...

Age
11 to 14
Challenge level
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Think of a number and follow my instructions. Tell me your answer, and I'll tell you what you started with! Can you explain how I know?
Perimeter Expressions
problem
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Perimeter Expressions

Age
11 to 14
Challenge level
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Create some shapes by combining two or more rectangles. What can you say about the areas and perimeters of the shapes you can make?
Calendar Capers
problem

Calendar Capers

Age
11 to 14
Challenge level
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Choose any three by three square of dates on a calendar page...
Folded A4
problem

Folded A4

Age
11 to 14
Challenge level
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What shapes can be made by folding an A4 sheet of paper only once?
Reflecting Lines
problem
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Reflecting Lines

Age
11 to 14
Challenge level
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Investigate what happens to the equations of different lines when you reflect them in one of the axes. Try to predict what will happen. Explain your findings.
Picturing Triangular Numbers
problem
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Picturing Triangular Numbers

Age
11 to 14
Challenge level
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Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Picturing Square Numbers
problem
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Picturing Square Numbers

Age
11 to 14
Challenge level
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Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Add to 200
problem
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Add to 200

Age
11 to 14
Challenge level
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By selecting digits for an addition grid, what targets can you make?
Days and Dates
problem
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Days and Dates

Age
11 to 14
Challenge level
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Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
How far does it move?
problem
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How far does it move?

Age
11 to 14
Challenge level
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Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the dot affects the distance it travels at each stage.

Translating Lines
problem
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Translating Lines

Age
11 to 14
Challenge level
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Investigate what happens to the equation of different lines when you translate them. Try to predict what will happen. Explain your findings.
What numbers can we make?
problem
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What numbers can we make?

Age
11 to 14
Challenge level
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Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

Multiple Surprises
problem
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Multiple Surprises

Age
11 to 16
Challenge level
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Sequences of multiples keep cropping up...
Semi-regular Tessellations
problem
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Semi-regular Tessellations

Age
11 to 16
Challenge level
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Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?
Growing Surprises
problem
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Growing Surprises

Age
11 to 14
Challenge level
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Can you find the connections between linear and quadratic patterns?
Diminishing Returns
problem
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Diminishing Returns

Age
11 to 14
Challenge level
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How much of the square is coloured blue? How will the pattern continue?
Polygon Rings
problem
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Polygon Rings

Age
11 to 14
Challenge level
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Join pentagons together edge to edge. Will they form a ring?
Sieve of Eratosthenes
problem
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Sieve of Eratosthenes

Age
11 to 14
Challenge level
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Follow this recipe for sieving numbers and see what interesting patterns emerge.
Island Hopping
problem

Island Hopping

Age
11 to 14
Challenge level
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What is the smallest number of ferry trips that Neda needs to take to visit all four islands and return to the mainland?
Odds, Evens and More Evens
problem
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Odds, Evens and More Evens

Age
11 to 14
Challenge level
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Alison, Bernard and Charlie have been exploring sequences of odd and even numbers, which raise some intriguing questions...

Polygon Pictures
problem
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Polygon Pictures

Age
11 to 14
Challenge level
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Can you work out how these polygon pictures were drawn, and use that to figure out their angles?
Square It
problem
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Square It

Age
11 to 16
Challenge level
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Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

Round and round and round
problem
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Round and round and round

Age
11 to 14
Challenge level
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Where will the point stop after it has turned through 30 000 degrees? I took out my calculator and typed 30 000 ÷ 360. How did this help?
Squares in a Square
problem

Squares in a Square

Age
11 to 14
Challenge level
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In the diagram, the small squares are all the same size. What fraction of the large square is shaded?
Mixing Lemonade
problem
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Mixing Lemonade

Age
11 to 14
Challenge level
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Can you work out which drink has the stronger flavour?

Fence it
problem
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Fence it

Age
11 to 14
Challenge level
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If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Constructing Triangles
problem
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Constructing Triangles

Age
11 to 14
Challenge level
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Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?
Daniel's Star
problem

Daniel's Star

Age
11 to 14
Challenge level
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A solid 'star' shape is created. How many faces does it have?
Isosceles Triangles
problem
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Isosceles Triangles

Age
11 to 14
Challenge level
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Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
Hidden Squares
problem
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Hidden Squares

Age
11 to 14
Challenge level
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Can you find the squares hidden on these coordinate grids?
Painted Octahedron
problem

Painted Octahedron

Age
11 to 14
Challenge level
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What is the smallest number of colours needed to paint the faces of a regular octahedron so that no adjacent faces are the same colour?
Frogs
problem
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Frogs

Age
11 to 14
Challenge level
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How many moves does it take to swap over some red and blue frogs? Do you have a method?
Soma Surface
problem

Soma Surface

Age
11 to 14
Challenge level
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What is the surface area of the solid shown?
Reflecting Squarely
problem
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Reflecting Squarely

Age
11 to 14
Challenge level
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In how many ways can you fit all three pieces together to make shapes with line symmetry?
Bishop's Paradise
problem

Bishop's Paradise

Age
11 to 14
Challenge level
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Weekly Problem 37 - 2013
Which of the statements about diagonals of polygons is false?
Fractions Rectangle
problem
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Fractions Rectangle

Age
11 to 14
Challenge level
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The large rectangle is divided into a series of smaller quadrilaterals and triangles. Can you untangle what fractional part is represented by each of the ten numbered shapes?
Odds and Evens
problem
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Odds and Evens

Age
11 to 14
Challenge level
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Are these games fair? How can you tell?
Potatoes
problem

Potatoes

Age
11 to 14
Challenge level
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Weekly Problem 19 - 2009
When I looked at the greengrocer's window I saw a sign. When I went in and looked from the other side, what did I see?
Printing Error
problem

Printing Error

Age
11 to 14
Challenge level
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Every third page number in this book has been omitted. Can you work out what number will be on the last page?
Shady Symmetry
problem
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Shady Symmetry

Age
11 to 14
Challenge level
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How many different symmetrical shapes can you make by shading triangles or squares?
Reflected Back
problem

Reflected Back

Age
11 to 14
Challenge level
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Imagine reflecting the letter P in all three sides of a triangle in turn. What is the final result?
Searching for mean(ing)
problem
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Searching for mean(ing)

Age
11 to 16
Challenge level
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If you have a large supply of 3kg and 8kg weights, how many of each would you need for the average (mean) of the weights to be 6kg?
Rolling Around
problem
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Rolling Around

Age
11 to 14
Challenge level
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A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?
Quadratic Patterns
problem
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Quadratic Patterns

Age
11 to 14
Challenge level
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Surprising numerical patterns can be explained using algebra and diagrams...
Rhombus It
problem
Favourite

Rhombus It

Age
11 to 16
Challenge level
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Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a rhombus.

Always a multiple?
problem
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Always a multiple?

Age
11 to 14
Challenge level
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Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...
An Unusual Shape
problem
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An Unusual Shape

Age
11 to 14
Challenge level
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Can you maximise the area available to a grazing goat?
Cuboid challenge
problem
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Cuboid challenge

Age
11 to 16
Challenge level
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What's the largest volume of box you can make from a square of paper?

Quadrilaterals in a Square
problem
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Quadrilaterals in a Square

Age
11 to 14
Challenge level
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What's special about the area of quadrilaterals drawn in a square?
Impossibilities
problem
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Impossibilities

Age
11 to 14
Challenge level
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Just because a problem is impossible doesn't mean it's difficult...
More Number Pyramids
problem
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More Number Pyramids

Age
11 to 14
Challenge level
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When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Reversals
problem
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Reversals

Age
11 to 14
Challenge level
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Where should you start, if you want to finish back where you started?
Same Answer
problem
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Same Answer

Age
11 to 14
Challenge level
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Aisha's division and subtraction calculations both gave the same answer! Can you find some more examples?
Shear Magic
problem
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Shear Magic

Age
11 to 14
Challenge level
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Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?
Opposite vertices
problem
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Opposite vertices

Age
11 to 14
Challenge level
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Can you recreate squares and rhombuses if you are only given a side or a diagonal?

11x11 square
problem

11x11 square

Age
11 to 16
Challenge level
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Here's a neat trick you can do with an 11 by 11 square...
Speeding up, slowing down
problem
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Speeding up, slowing down

Age
11 to 14
Challenge level
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Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the dot affects its speed at each stage.

Pythagoras Proofs
problem
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Pythagoras Proofs

Age
11 to 16
Challenge level
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Can you make sense of these three proofs of Pythagoras' Theorem?

Seven Squares - Group-worthy Task
problem
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Seven Squares - Group-worthy Task

Age
11 to 14
Challenge level
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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?
Fill Me Up
problem
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Fill Me Up

Age
11 to 14
Challenge level
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Can you sketch graphs to show how the height of water changes in different containers as they are filled?

Counting Factors
problem
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Counting Factors

Age
11 to 14
Challenge level
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Is there an efficient way to work out how many factors a large number has?
Coordinate Patterns
problem
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Coordinate Patterns

Age
11 to 14
Challenge level
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Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?
Squares in rectangles
problem
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Squares in rectangles

Age
11 to 14
Challenge level
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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?
Tower of Hanoi
problem
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Tower of Hanoi

Age
11 to 14
Challenge level
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The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
Legs Eleven
problem
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Legs Eleven

Age
11 to 14
Challenge level
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Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11?
Route to infinity
problem
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Route to infinity

Age
11 to 14
Challenge level
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Can you describe this route to infinity? Where will the arrows take you next?
On the Edge
problem
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On the Edge

Age
11 to 14
Challenge level
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If you move the tiles around, can you make squares with different coloured edges?
Triangle Numbers
problem
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Triangle Numbers

Age
11 to 14
Challenge level
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Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?
Funny Factorisation
problem
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Funny Factorisation

Age
11 to 16
Challenge level
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Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?
Triathlon and Fitness
problem
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Triathlon and Fitness

Age
11 to 14
Challenge level
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The triathlon is a physically gruelling challenge. Can you work out which athlete burnt the most calories?
Doubly Symmetric
problem

Doubly Symmetric

Age
11 to 14
Challenge level
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What is the smallest number of additional squares that must be shaded so that this figure has at least one line of symmetry and rotational symmetry of order 2?
Don't Be Late
problem

Don't Be Late

Age
11 to 14
Challenge level
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Mary is driving to Birmingham Airport. Using her average speed for the entire journey, find how long her journey took.
Marbles in a box
problem
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Marbles in a box

Age
11 to 16
Challenge level
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How many winning lines can you make in a three-dimensional version of noughts and crosses?
Integral Polygons
problem

Integral Polygons

Age
11 to 14
Challenge level
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Each interior angle of a particular polygon is an obtuse angle which is a whole number of degrees. What is the greatest number of sides the polygon could have?
Blockupied
problem

Blockupied

Age
11 to 14
Challenge level
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A 1x2x3 block is placed on an 8x8 board and rolled several times.... How many squares has it occupied altogether?
Hamiltonian Cube
problem
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Hamiltonian Cube

Age
11 to 16
Challenge level
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Weekly Problem 36 - 2007
Find the length along the shortest path passing through certain points on the cube.
Seven Squares
problem
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Seven Squares

Age
11 to 14
Challenge level
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Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
Same Face
problem

Same Face

Age
11 to 14
Challenge level
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A cube is rolled on a plane, landing on the squares in the order shown. Which two positions had the same face of the cube touching the surface?
Think of Two Numbers
problem
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Think of Two Numbers

Age
11 to 14
Challenge level
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Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
Square coordinates
problem
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Square coordinates

Age
11 to 14
Challenge level
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A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?

Stars
problem
Favourite

Stars

Age
11 to 14
Challenge level
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Can you work out what step size to take to ensure you visit all the dots on the circle?
Night Watchmen
problem

Night Watchmen

Age
11 to 14
Challenge level
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Grannie's watch gains 30 minutes every hour, whilst Grandpa's watch loses 30 minutes every hour. What is the correct time when their watches next agree?
Turning N Over
problem

Turning N Over

Age
11 to 14
Challenge level
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A card with the letter N on it is rotated through two different axes. What does the card look like at the end?
Adjacent Factors
problem

Adjacent Factors

Age
11 to 14
Challenge level
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Two numbers can be placed adjacent if one of them divides the other. Using only $1,...,10$, can you write the longest such list?
The Farmers' Field Boundary
problem
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The Farmers' Field Boundary

Age
11 to 14
Challenge level
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The farmers want to redraw their field boundary but keep the area the same. Can you advise them?
Shapely pairs
problem
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Shapely pairs

Age
11 to 14
Challenge level
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A game in which players take it in turns to turn up two cards. If they can draw a triangle which satisfies both properties they win the pair of cards. And a few challenging questions to follow...

Fifty Coins
problem

Fifty Coins

Age
11 to 14
Challenge level
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Cheryl finds a bag of coins. Can you work out how many more 5p coins than 2p coins are in the bag?
Alison's quilt
problem
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Alison's quilt

Age
11 to 14
Challenge level
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Nine squares are fitted together to form a rectangle. Can you find its dimensions?
Mixing Paints
problem
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Mixing Paints

Age
11 to 14
Challenge level
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Can you work out how to produce different shades of pink paint?
Product and Sum
problem

Product and Sum

Age
11 to 14
Challenge level
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When Jim rolled some dice, the scores had the same product and sum. How many dice did Jim roll?