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Broad Topics >

Using, Applying and Reasoning about Mathematics > Mathematical induction

##### Age 14 to 16 Challenge Level:

A walk is made up of diagonal steps from left to right, starting at
the origin and ending on the x-axis. How many paths are there for 4
steps, for 6 steps, for 8 steps?

##### Age 14 to 16

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

##### 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.

##### Age 7 to 18 Challenge Level:

Libby Jared helped to set up NRICH and this is one of her favourite
problems. It's a problem suitable for a wide age range and best
tackled practically.

##### Age 16 to 18 Challenge Level:

With n people anywhere in a field each shoots a water pistol at the
nearest person. In general who gets wet? What difference does it
make if n is odd or even?

##### Age 16 to 18 Challenge Level:

By proving these particular identities, prove the existence of general cases.

##### Age 16 to 18 Challenge Level:

Find and explain a short and neat proof that 5^(2n+1) + 11^(2n+1) +
17^(2n+1) is divisible by 33 for every non negative integer n.

##### Age 16 to 18 Challenge Level:

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

##### Age 16 to 18 Challenge Level:

When is $7^n + 3^n$ a multiple of 10? Can you prove the result by two different methods?

##### Age 16 to 18 Challenge Level:

Farey sequences are lists of fractions in ascending order of magnitude. Can you prove that in every Farey sequence there is a special relationship between Farey neighbours?

##### Age 16 to 18 Challenge Level:

Add powers of 3 and powers of 7 and get multiples of 11.

##### Age 14 to 18 Challenge Level:

Find the values of n for which 1^n + 8^n - 3^n - 6^n is divisible
by 6.

##### Age 16 to 18 Challenge Level:

Explore the hyperbolic functions sinh and cosh using what you know about the exponential function.

##### Age 16 to 18

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

##### 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)?

##### Age 16 to 18 Challenge Level:

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

##### Age 16 to 18 Challenge Level:

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

##### Age 16 to 18 Short Challenge Level:

Investigate Farey sequences of ratios of Fibonacci numbers.

##### Age 16 to 18 Challenge Level:

In the limit you get the sum of an infinite geometric series. What
about an infinite product (1+x)(1+x^2)(1+x^4)... ?

##### Age 16 to 18 Challenge Level:

Bricks are 20cm long and 10cm high. How high could an arch be built
without mortar on a flat horizontal surface, to overhang by 1
metre? How big an overhang is it possible to make like this?

##### Age 16 to 18 Challenge Level:

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

##### Age 16 to 18 Challenge Level:

Find the link between a sequence of continued fractions and the
ratio of succesive Fibonacci numbers.