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

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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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Explore this how this program produces the sequences it does. What are you controlling when you change the values of the variables?

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Pentagram Pylons - can you elegantly recreate them? Or, the European flag in LOGO - what poses the greater problem?

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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.

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Four friends must cross a bridge. How can they all cross it in just 17 minutes?

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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.

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

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This Sudoku problem consists of a pair of linked standard Suduko puzzles each with some starting digits

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

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

This is about a fiendishly difficult jigsaw and how to solve it using a computer program.

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

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

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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.

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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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?

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

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

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Remember that you want someone following behind you to see where you went. Can yo work out how these patterns were created and recreate them?

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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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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.

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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.

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

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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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Just four procedures were used to produce a design. How was it done? Can you be systematic and elegant so that someone can follow your logic?

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

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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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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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This Sudoku, based on differences. Using the one clue number can you find the solution?

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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?

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

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

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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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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Use the differences to find the solution to this Sudoku.

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Time for a little mathemagic! Choose any five cards from a pack and show four of them to your partner. How can they work out the fifth?

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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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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?