Toughnuts

 

Try these unsolved problems

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Turning

Stage:1 Challenge Level:Challenge Level:1

Use your mouse to move the red and green parts of this disc. Can you make images which show the turnings described?

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Tug of War

Stage:1 Challenge Level:Challenge Level:1

Can you use the numbers on the dice to reach your end of the number line before your partner beats you?

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Caterpillars

Stage:1 Challenge Level:Challenge Level:2Challenge Level:2

These caterpillars have 16 parts. What different shapes do they make if each part lies in the small squares of a 4 by 4 square?

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Overlaps

Stage:1 Challenge Level:Challenge Level:2Challenge Level:2

What does the overlap of these two shapes look like? Try picturing it in your head and then use the interactivity to test your prediction.

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Inside Triangles

Stage:1 Challenge Level:Challenge Level:3Challenge Level:3Challenge Level:3

How many different triangles can you draw on the dotty grid which each have one dot in the middle?

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Stop the Clock

Stage:1 Challenge Level:Challenge Level:3Challenge Level:3Challenge Level:3

This is a game for two players. Can you find out how to be the first to get to 12 o'clock?

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Tables Without Tens

Stage:2 Challenge Level:Challenge Level:1

Investigate and explain the patterns that you see from recording just the units digits of numbers in the times tables.

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More Building with Cubes

Stage:2 Challenge Level:Challenge Level:1

Here are more buildings to picture in your mind's eye. Watch out - they become quite complicated!

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Tug Harder!

Stage:2 Challenge Level:Challenge Level:1

In this game, you can add, subtract, multiply or divide the numbers on the dice. Which will you do so that you get to the end of the number line first?

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Putting Two and Two Together

Stage:2 Challenge Level:Challenge Level:1

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

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Tessellating Transformations

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

Can you find out how the 6-triangle shape is transformed in these tessellations? Will the tessellations go on for ever? Why or why not?

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Alien Counting

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

Investigate the different ways these aliens count in this challenge. You could start by thinking about how each of them would write our number 7.

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Overlapping Again

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

What shape is the overlap when you slide one of these shapes half way across another? Can you picture it in your head? Use the interactivity to check your visualisation.

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Shaping Up

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

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

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How Random!

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

Explore this interactivity and see if you can work out what it does. Could you use it to estimate the area of a shape?

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Tetrahedron Faces

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

One face of a regular tetrahedron is painted blue and each of the remaining faces are painted using one of the colours red, green or yellow. How many different possibilities are there?

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A Day with Grandpa

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

Grandpa was measuring a rug using yards, feet and inches. Can you help William to work out its area?

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Mobile Numbers

Stage:1 and 2 Challenge Level:Challenge Level:1

In this investigation, you are challenged to make mobile phone numbers which are easy to remember. What happens if you make a sequence adding 2 each time?

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Double Digit

Stage:3 Challenge Level:Challenge Level:1

Choose two digits and arrange them to make two double-digit numbers. Now add your double-digit numbers. Now add your single digit numbers. Divide your double-digit answer by your single-digit answer. . . .

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Notes on a Triangle

Stage:3 Challenge Level:Challenge Level:1

Can you describe what happens in this film?

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Bow Tie

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

Show how this pentagonal tile can be used to tile the plane and describe the transformations which map this pentagon to its images in the tiling.

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Drilling Many Cubes

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

A useful visualising exercise which offers opportunities for discussion and generalising, and which could be used for thinking about the formulae needed for generating the results on a spreadsheet.

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Playing Connect Three

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

In this game the winner is the first to complete a row of three. Are some squares easier to land on than others?

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Conway's Chequerboard Army

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

Here is a solitaire type environment for you to experiment with. Which targets can you reach?

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Which Spinners?

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

Which spinners were used to generate these frequency charts?

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Where Is the Dot?

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

A dot starts at the point (1,0) and turns anticlockwise. Can you estimate the height of the dot after it has turned through 45 degrees? Can you calculate its height?

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Turning Tangles

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

Look carefully at the video of a tangle and explain what's happening.

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First Connect Three

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

The idea of this game is to add or subtract the two numbers on the dice and cover the result on the grid, trying to get a line of three. Are there some numbers that are good to aim for?

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Twizzle Arithmetic

Stage:4 Challenge Level:Challenge Level:1

Arrow arithmetic, but with a twist.

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Twizzles Venture Forth

Stage:4 Challenge Level:Challenge Level:1

Where we follow twizzles to places that no number has been before.

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A Scale for the Solar System

Stage:4 Challenge Level:Challenge Level:1

The Earth is further from the Sun than Venus, but how much further? Twice as far? Ten times?

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The Perforated Cube

Stage:4 Challenge Level:Challenge Level:1

A cube is made from smaller cubes, 5 by 5 by 5, then some of those cubes are removed. Can you make the specified shapes, and what is the most and least number of cubes required ?

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Negatively Triangular

Stage:4 Challenge Level:Challenge Level:1

How many intersections do you expect from four straight lines ? Which three lines enclose a triangle with negative co-ordinates for every point ?

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Stretching Fractions

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

Imagine a strip with a mark somewhere along it. Fold it in the middle so that the bottom reaches back to the top. Stetch it out to match the original length. Now where's the mark?

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Triangle in a Triangle

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

Can you work out the fraction of the original triangle that is covered by the inner triangle?

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The Line and Its Strange Pair

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

In the diagram the point P' can move to different places along the dotted line. Each position P' takes will fix a corresponding position for P. If P' moves along a straight line what does P do ?

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Twizzle Wind Up

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

A loopy exploration of z^2+1=0 (z squared plus one) with an eye on winding numbers. Try not to get dizzy!

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Eight Ratios

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

Two perpendicular lines lie across each other and the end points are joined to form a quadrilateral. Eight ratios are defined, three are given but five need to be found.

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Gutter

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

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

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The Invertible Trefoil

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

When is a knot invertible ?

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The Fire-fighter's Car Keys

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

A fire-fighter needs to fill a bucket of water from the river and take it to a fire. What is the best point on the river bank for the fire-fighter to fill the bucket ?.

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The Legacy

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

Your school has been left a million pounds in the will of an ex- pupil. What model of investment and spending would you use in order to ensure the best return on the money?

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Zeller's Birthday

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

What day of the week were you born on? Do you know? Here's a way to find out.

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Introducing Distributions

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

When five dice are rolled together which do you expect to see more often, no sixes or all sixes ?

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Distribution Differences

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

How could you compare different situation where something random happens ? What sort of things might be the same ? What might be different ?

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Tri-split

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

A point P is selected anywhere inside an equilateral triangle. What can you say about the sum of the perpendicular distances from P to the sides of the triangle? Can you prove your conjecture?

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The Rescaled Map

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

We use statistics to give ourselves an informed view on a subject of interest. This problem explores how to scale countries on a map to represent characteristics other than land area.

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Mapping the Wandering Circle

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

In the diagram the point P can move to different places around the dotted circle. Each position P takes will fix a corresponding position for P'. As P moves around on that circle what will P' do?

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Wedge on Wedge

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

Two right-angled triangles are connected together as part of a structure. An object is dropped from the top of the green triangle where does it pass the base of the blue triangle?

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Twizzle Twists

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

Make the twizzle twist on its spot and so work out the hidden link.

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Dating Made Easier

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

If a sum invested gains 10% each year how long before it has doubled its value?

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Modular Knights

Stage:5 Challenge Level:Challenge Level:1

Try to move the knight to visit each square once and return to the starting point. Move either 2 steps one way and one perpendicular (as in chess) or generalise to a steps one way and b the other.

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Epidemic Modelling

Stage:5 Challenge Level:Challenge Level:1

Use the computer to model an epidemic. Try out public health policies to control the spread of the epidemic, to minimise the number of sick days and deaths.

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Overarch 1

Stage:5 Challenge Level:Challenge Level:1

This short question asks if you can work out the most precarious way to balance four tiles.

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Universal Time, Mass, Length

Stage:5 Short Challenge Level:Challenge Level:1

Can you work out the natural time scale for the universe?

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The Monte Carlo Method

Stage:5 Challenge Level:Challenge Level:1

Estimate areas using random grids

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Random Squares

Stage:5 Challenge Level:Challenge Level:1

What is a random pattern?

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Dam Busters 2

Stage:5 Challenge Level:Challenge Level:2Challenge Level:2

Can you work out which of the equations models a bouncing bomb? Will you be able to hit the target?

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More Bridge Building

Stage:5 Challenge Level:Challenge Level:2Challenge Level:2

Which parts of these framework bridges are in tension and which parts are in compression?

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Impossible Square?

Stage:5 Challenge Level:Challenge Level:2Challenge Level:2

Can you make a square from these triangles?

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It's Only a Minus Sign

Stage:5 Challenge Level:Challenge Level:2Challenge Level:2

Solve these differential equations to see how a minus sign can change the answer

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Equation Matcher

Stage:5 Challenge Level:Challenge Level:2Challenge Level:2

Match the descriptions of physical processes to these differential equations.

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Road Maker 2

Stage:5 Short Challenge Level:Challenge Level:2Challenge Level:2

Can you work out where the blue-and-red brick roads end?

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Scale Invariance

Stage:5 Challenge Level:Challenge Level:2Challenge Level:2

By exploring the concept of scale invariance, find the probability that a random piece of real data begins with a 1.

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Escape from Planet Earth

Stage:5 Challenge Level:Challenge Level:2Challenge Level:2

How fast would you have to throw a ball upwards so that it would never land?

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Vecten

Stage:5 Challenge Level:Challenge Level:3Challenge Level:3Challenge Level:3

Join in this ongoing research. Build squares on the sides of a triangle, join the outer vertices forming hexagons, build further rings of squares and quadrilaterals, investigate.

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Quaternions and Reflections

Stage:5 Challenge Level:Challenge Level:3Challenge Level:3Challenge Level:3

See how 4 dimensional quaternions involve vectors in 3-space and how the quaternion function F(v) = nvn gives a simple algebraic method of working with reflections in planes in 3-space.

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Cheese Cutting

Stage:5 Challenge Level:Challenge Level:3Challenge Level:3Challenge Level:3

In this problem we see how many pieces we can cut a cube of cheese into using a limited number of slices. How many pieces will you be able to make?

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Earth Orbit

Stage:5 Challenge Level:Challenge Level:3Challenge Level:3Challenge Level:3

Follow in the steps of Newton and find the path that the earth follows around the sun.

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Circular Circuitry

Stage:5 Challenge Level:Challenge Level:3Challenge Level:3Challenge Level:3

What will happen when you switch on these circular circuits?

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Random Inequalities

Stage:5 Challenge Level:Challenge Level:3Challenge Level:3Challenge Level:3

Can you build a distribution with the maximum theoretical spread?

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Giant Holly Leaf

Stage:4 and 5 Challenge Level:Challenge Level:1

You are asked to find the perimeter and area of a 'holly leaf' Like a real holly leaf, it will not ie flat (it has negative curvature) and what looks like a circle has circumference greater. . . .

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Root Hunter

Stage:4 and 5 Challenge Level:Challenge Level:1

In this short problem, try to find the location of the roots of some unusual functions by finding where they change sign.

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Impossible Triangles?

Stage:4 and 5 Challenge Level:Challenge Level:1

Which of these triangular jigsaws are impossible to finish?

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Data Matching

Stage:4 and 5 Short Challenge Level:Challenge Level:1

Use your skill and judgement to match the sets of random data.

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Pentagon from 1926

Stage:4 and 5 Challenge Level:Challenge Level:2Challenge Level:2

Find the vertices of a pentagon given the midpoints of its sides.

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Triple Decker Sandwich

Stage:4 and 5 Challenge Level:Challenge Level:3Challenge Level:3Challenge Level:3

There are 150 8-sandwiches like 6 1 5 1 8 4 7 6 5 2 4 3 2 8 7 3 with 1 number between the 1's, 2 between the 2's etc. Can you find sandwiches where each digit occurs three times rather than twice?

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Balance Power

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

Using balancing scales what is the least number of weights needed to weigh all integer masses from 1 to 1000? Placing some of the weights in the same pan as the object how many are needed?