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

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

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

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.

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

Investigate Farey sequences of ratios of Fibonacci numbers.

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 ?

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.

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

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?

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

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 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.

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

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.

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

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?

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