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Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?

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

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Make some loops out of regular hexagons. What rules can you discover?

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Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

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Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

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Show that all pentagonal numbers are one third of a triangular number.

Alf Coles writes about how he tries to create 'spaces for exploration' for the students in his classrooms.

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Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...

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Explore the relationship between quadratic functions and their graphs.

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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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Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

This article for teachers discusses examples of problems in which there is no obvious method but in which children can be encouraged to think deeply about the context and extend their ability to. . . .

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I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

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Can you find a rule which relates triangular numbers to square numbers?

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Can you find a rule which connects consecutive triangular numbers?

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In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?

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Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.

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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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What's the greatest number of sides a polygon on a dotty grid could have?

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Explore the relationship between simple linear functions and their graphs.

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Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

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Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

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Drawing a triangle is not always as easy as you might think!

Find out about Magic Squares in this article written for students. Why are they magic?!

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Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

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Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?

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The points P, Q, R and S are the midpoints of the edges of a non-convex quadrilateral.What do you notice about the quadrilateral PQRS and its area?

An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.