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Broad Topics > Secondary processes > PM - Reasoning, Justifying. Convincing. Proof.

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Seven Squares

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

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

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Stats Statements

Stage: 5 Challenge Level: Challenge Level:1

Are these statistical statements sometimes, always or never true? Or it is impossible to say?

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Route to Infinity

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

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

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Odds and Evens

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

Is this a fair game? How many ways are there of creating a fair game by adding odd and even numbers?

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Arithmagons

Stage: 3 Challenge Level: Challenge Level:1

Can you find the values at the vertices when you know the values on the edges?

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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. How many of the small cubes will have yellow paint on their faces?

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Tilted Squares

Stage: 3 Challenge Level: Challenge Level:1

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

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Number Pyramids

Stage: 3 Challenge Level: Challenge Level:1

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

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Take Three from Five

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

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

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What's it Worth?

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

There are lots of different methods to find out what the shapes are worth - how many can you find?

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1 Step 2 Step

Stage: 3 Challenge Level: Challenge Level:1

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

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Attractive Tablecloths

Stage: 4 Challenge Level: Challenge Level:1

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

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Marbles in a Box

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

In a three-dimensional version of noughts and crosses, how many winning lines can you make?

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What's Possible?

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

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

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Summing Consecutive Numbers

Stage: 3 Challenge Level: Challenge Level:1

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

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Polynomial Interpolation

Stage: 5 Challenge Level: Challenge Level:1

Can you fit polynomials through these points?

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Magic Letters

Stage: 3 Challenge Level: Challenge Level:1

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

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What Numbers Can We Make?

Stage: 3 Challenge Level: Challenge Level:1

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

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Diminishing Returns

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

In this problem, we have created a pattern from smaller and smaller squares. If we carried on the pattern forever, what proportion of the image would be coloured blue?

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Prime Sequences

Stage: 5 Challenge Level: Challenge Level:1

This group tasks allows you to search for arithmetic progressions in the prime numbers. How many of the challenges will you discover for yourself?

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Integration Matcher

Stage: 5 Challenge Level: Challenge Level:1

Match the charts of these functions to the charts of their integrals.

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Dodgy Proofs

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

These proofs are wrong. Can you see why?

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Impossible Triangles?

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

Which of these triangular jigsaws are impossible to finish?

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Impossible Square?

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

Can you make a square from these triangles?

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What's a Group?

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

Explore the properties of some groups such as: The set of all real numbers excluding -1 together with the operation x*y = xy + x + y. Find the identity and the inverse of the element x.