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### Arithmagons

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

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### Take Three From Five

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

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### More Twisting and Turning

It would be nice to have a strategy for disentangling any tangled ropes...

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### Differences

Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?

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### Pair Products

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

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### Speeding boats

Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?

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### Circles in quadrilaterals

Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

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### A little light thinking

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

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### Which is cheaper?

When I park my car in Mathstown, there are two car parks to choose from. Can you help me to decide which one to use?

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### Plus Minus

Can you explain the surprising results Jo found when she calculated
the difference between square numbers?

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### Of all the areas

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

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### Fair Shares?

A mother wants to share a sum of money by giving each of her
children in turn a lump sum plus a fraction of the remainder. How
can she do this in order to share the money out equally?

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### What's Possible?

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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### Attractive Tablecloths

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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### Pick's Theorem

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

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### Painted Cube

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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### Multiplication square

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

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### Perpendicular lines

Position the lines so that they are perpendicular to each other. What can you say about the equations of perpendicular lines?

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### For richer for poorer

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

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### At right angles

Can you decide whether two lines are perpendicular or not? Can you do this without drawing them?

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### Mystic Rose

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

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### Harmonic Triangle

Can you see how to build a harmonic triangle? Can you work out the next two rows?