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?

Peter Zimmerman from Mill Hill County High School in Barnet, London gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.

A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers?

An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum of two or more cubes.

Can you explain why a sequence of operations always gives you perfect squares?

Show that the arithmetic mean, geometric mean and harmonic mean of a and b can be the lengths of the sides of a right-angles triangle if and only if a = bx^3, where x is the Golden Ratio.

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...

Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.

An introduction to some beautiful results of Number Theory (a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions)

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.

What can you say about the common difference of an AP where every term is prime?

How many noughts are at the end of these giant numbers?

By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

What is the largest number of intersection points that a triangle and a quadrilateral can have?

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

We only need 7 numbers for modulus (or clock) arithmetic mod 7 including working with fractions. Explore how to divide numbers and write fractions in modulus arithemtic.

Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.

Can you correctly order the steps in the proof of the formula for the sum of a geometric series?

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

Can you rearrange the cards to make a series of correct mathematical statements?

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

When is it impossible to make number sandwiches?

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate. . . .

Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.

Explore a number pattern which has the same symmetries in different bases.

Show that for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity.

The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . .

We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.

The knight's move on a chess board is 2 steps in one direction and one step in the other direction. Prove that a knight cannot visit every square on the board once and only (a tour) on a 2 by n board. . . .

The sums of the squares of three related numbers is also a perfect square - can you explain why?

Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

A connected graph is a graph in which we can get from any vertex to any other by travelling along the edges. A tree is a connected graph with no closed circuits (or loops. Prove that every tree has. . . .

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

Kyle and his teacher disagree about his test score - who is right?

Can you work out where the blue-and-red brick roads end?