Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.

Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2

Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.

Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?

Show there are exactly 12 magic labellings of the Magic W using the numbers 1 to 9. Prove that for every labelling with a magic total T there is a corresponding labelling with a magic total 30-T.

A pair of Sudokus with lots in common. In fact they are the same problem but rearranged. Can you find how they relate to solve them both?

The challenge is to find the values of the variables if you are to solve this Sudoku.

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

Find out about Magic Squares in this article written for students. Why are they magic?!

In this article, the NRICH team describe the process of selecting solutions for publication on the site.

A Sudoku based on clues that give the differences between adjacent cells.

A Latin square of order n is an array of n symbols in which each symbol occurs exactly once in each row and exactly once in each column.

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

Special clue numbers related to the difference between numbers in two adjacent cells and values of the stars in the "constellation" make this a doubly interesting problem.

Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest?

Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.

How many solutions can you find to this sum? Each of the different letters stands for a different number.

Imagine a stack of numbered cards with one on top. Discard the top, put the next card to the bottom and repeat continuously. Can you predict the last card?

Four small numbers give the clue to the contents of the four surrounding cells.

Mr Smith and Mr Jones are two maths teachers. By asking questions, the answers to which may be right or wrong, Mr Jones is able to find the number of the house Mr Smith lives in... Or not!

in how many ways can you place the numbers 1, 2, 3 … 9 in the nine regions of the Olympic Emblem (5 overlapping circles) so that the amount in each ring is the same?

In this Sudoku, there are three coloured "islands" in the 9x9 grid. Within each "island" EVERY group of nine cells that form a 3x3 square must contain the numbers 1 through 9.

You are given the Lowest Common Multiples of sets of digits. Find the digits and then solve the Sudoku.

Use the differences to find the solution to this Sudoku.

Four numbers on an intersection that need to be placed in the surrounding cells. That is all you need to know to solve this sudoku.

Ben, Jack and Emma passed counters to each other and ended with the same number of counters. How many did they start with?

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

This is a variation of sudoku which contains a set of special clue-numbers. Each set of 4 small digits stands for the numbers in the four cells of the grid adjacent to this set.

It is possible to identify a particular card out of a pack of 15 with the use of some mathematical reasoning. What is this reasoning and can it be applied to other numbers of cards?

The puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times9$ grid.

A pair of Sudoku puzzles that together lead to a complete solution.

Solve the equations to identify the clue numbers in this Sudoku problem.

Label the joints and legs of these graph theory caterpillars so that the vertex sums are all equal.

A function pyramid is a structure where each entry in the pyramid is determined by the two entries below it. Can you figure out how the pyramid is generated?

Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.

This sudoku requires you to have "double vision" - two Sudoku's for the price of one

This Sudoku requires you to do some working backwards before working forwards.

This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.

What is the smallest perfect square that ends with the four digits 9009?

How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?

Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.

Can you swap the black knights with the white knights in the minimum number of moves?

A particular technique for solving Sudoku puzzles, known as "naked pair", is explained in this easy-to-read article.

The clues for this Sudoku are the product of the numbers in adjacent squares.