Toughnuts

Try these unsolved problems

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Teddy Bear Line-up

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

What is the least number of moves you can take to rearrange the bears so that no bear is next to a bear of the same colour?

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Clock Face

Stage: 2 Challenge Level: Challenge Level:1

Where can you draw a line on a clock face so that the numbers on both sides have the same total?

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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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Locate the Lion's Lair

Stage: 2 Challenge Level: Challenge Level:1

Use the sightings of the lion to guess the location of its lair.

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How Random!

Stage: 2 Challenge Level: Challenge Level:2 Challenge 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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Largest Product

Stage: 3 Challenge Level: Challenge Level:1

Which set of numbers that add to 10 have the largest product?

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In the Bag

Stage: 3 Challenge Level: Challenge Level:1

Can you guess the colours of the 10 marbles in the bag? Can you develop an effective strategy for reaching 1000 points in the least number of rounds?

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Reaction Timer

Stage: 3 Challenge Level: Challenge Level:1

What are your reactions like and what affects them?

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Reflecting Lines

Stage: 3 Challenge Level: Challenge Level:1

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.

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Translating Lines

Stage: 3 Challenge Level: Challenge Level:1

Investigate what happens to the equation of different lines when you translate them. Try to predict what will happen. Explain your findings.

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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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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.

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Playing Connect Three

Stage: 3 Challenge Level: Challenge Level:2 Challenge 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:2 Challenge 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:2 Challenge 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:3 Challenge Level:3 Challenge 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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For Richer for Poorer

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

Charlie has moved between countries and the average income of both has increased. How can this be so?

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The Greedy Algorithm

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

The Egyptians expressed all fractions as the sum of different unit fractions. The Greedy Algorithm might provide us with an efficient way of doing this.

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Surprising Transformations

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

I took the graph y=4x+7 and performed four transformations. Can you find the order in which I could have carried out the transformations?

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Cops and Robbers

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

Can you find a reliable strategy for choosing coordinates that will locate the robber in the minimum number of guesses?

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First Connect Three

Stage: 2 and 3 Challenge Level: Challenge Level:2 Challenge 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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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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Power Countdown

Stage: 4 Challenge Level: Challenge Level:1

In this twist on the well-known Countdown numbers game, use your knowledge of Powers and Roots to make a target.

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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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Contact

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

A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?

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Of All the Areas

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

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

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Take a Square

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

The diagonal of a square intersects the line joining one of the unused corners to the midpoint of the opposite side. What do you notice about the line segments produced?

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Gnomon Dimensions

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

These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.

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The Invertible Trefoil

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

When is a knot invertible ?

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Distribution Differences

Stage: 4 Challenge Level: Challenge Level:2 Challenge 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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Take a Square II

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

What fractions can you divide the diagonal of a square into by simple folding?

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Tri-split

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge 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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Giant Holly Leaf

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

Find the perimeter and area of a holly leaf that will not lie flat (it has negative curvature with 'circles' having circumference greater than 2πr).

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Pentagon

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

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

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Platonic Planet

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

Glarsynost the alien lives on a platonic planet whose shape is that of a perfect regular dodecahedron. Can you describe the shortest journey she can make that ensures she will see every part of. . . .

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Six Discs

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

Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?

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Circumspection

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

M is any point on the line AB. Squares of side length AM and MB are constructed and their circumcircles intersect at P (and M). Prove that the lines AD and BE produced pass through P.

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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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Into the Exponential Distribution

Stage: 5 Challenge Level: Challenge Level:1

Get into the exponential distribution through an exploration of its pdf.

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What's Your Mean?

Stage: 5 Challenge Level: Challenge Level:1

Can you work out the means of these distributions using numerical methods?

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Rain or Shine

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

Predict future weather using the probability that tomorrow is wet given today is wet and the probability that tomorrow is wet given that today is dry.

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More Bridge Building

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

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

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Spot the Difference

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

If you plot these graphs they may look the same, but are they?

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PCDF

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

When can a pdf and a cdf coincide?

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Into the Normal Distribution

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

Investigate the normal distribution

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Time to Evolve 2

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

How is the length of time between the birth of an animal and the birth of its great great ... great grandparent distributed?

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Road Maker 2

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

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

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Circular Circuitry

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

What will happen when you switch on these circular circuits?

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Random Inequalities

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

Can you build a distribution with the maximum theoretical spread?

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Pattern Recognition

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

When does a pattern start to exhibit structure? Can you crack the code used by the computer?

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Lion Hunting

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

A killer lion is causing devastation. From the locations of its reported activity, can you work out where its lair is located?