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

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

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

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

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?

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?

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

Investigate Farey sequences of ratios of Fibonacci numbers.

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.

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

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

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

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)... ?

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

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.

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?

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?

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.

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

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