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

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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?

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

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

Read this article to find out more about the inspiration for NRICH's game, Phiddlywinks.

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

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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.

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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.

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

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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.

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.

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Take three whole numbers. The differences between them give you three new numbers. Find the differences between the new numbers and keep repeating this. What happens?

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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.

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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.

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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?

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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?

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

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

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

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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.

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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.

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

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

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.

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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!

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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?

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This Sudoku puzzle can be solved with the help of small clue-numbers on the border lines between pairs of neighbouring squares of the grid.

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Given the products of diagonally opposite cells - can you complete this Sudoku?

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

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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.

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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?

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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.

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

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

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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?

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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.

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

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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?

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Can you use your powers of logic and deduction to work out the missing information in these sporty situations?

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Two sudokus in one. Challenge yourself to make the necessary connections.

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

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