Mathematical induction

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An introduction to mathematical induction

This article gives an introduction to mathematical induction, a powerful method of mathematical proof.
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Symmetric tangles

The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!
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Binomial coefficients

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

Age

16 to 18

Challenge level

Add powers of 3 and powers of 7 and get multiples of 11.
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OK! Now prove it

Age

16 to 18

Challenge level

Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?
problem
Growing

Age

16 to 18

Challenge level

Which is larger: (a) 1.000001^{1000000} or 2? (b) 100^{300} or 300! (i.e.factorial 300)
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Binary squares

Age

16 to 18

Challenge level

If a number N is expressed in binary by using only 'ones,' what can you say about its square (in binary)?
problem
Golden powers

Age

16 to 18

Challenge level

You add 1 to the golden ratio to get its square. How do you find higher powers?
problem
Counting binary ops

Age

14 to 16

Challenge level

How many ways can the terms in an ordered list be combined by repeating a single binary operation. Show that for 4 terms there are 5 cases and find the number of cases for 5 terms and 6 terms.
problem
Fibonacci fashion

Age

16 to 18

Challenge level

What have Fibonacci numbers to do with solutions of the quadratic equation x^2 - x - 1 = 0 ?