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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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An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

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Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

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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.

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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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Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

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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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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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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Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

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60 pieces and a challenge. What can you make and how many of the pieces can you use creating skeleton polyhedra?

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How many different symmetrical shapes can you make by shading triangles or squares?

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In this game you are challenged to gain more columns of lily pads than your opponent.

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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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This task encourages you to investigate the number of edging pieces and panes in different sized windows.

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Find the smallest whole number which, when mutiplied by 7, gives a product consisting entirely of ones.

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There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

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The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

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

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Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

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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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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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Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

There is a long tradition of creating mazes throughout history and across the world. This article gives details of mazes you can visit and those that you can tackle on paper.

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Can you arrange the digits 1, 1, 2, 2, 3 and 3 to make a Number Sandwich?

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A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

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If you take a three by three square on a 1-10 addition square and multiply the diagonally opposite numbers together, what is the difference between these products. Why?

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I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

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Investigate the different ways that fifteen schools could have given money in a charity fundraiser.

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Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?

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15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

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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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Rather than using the numbers 1-9, this sudoku uses the nine different letters used to make the words "Advent Calendar".

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You need to find the values of the stars before you can apply normal Sudoku rules.

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This pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find it?

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Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

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Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

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An extra constraint means this Sudoku requires you to think in diagonals as well as horizontal and vertical lines and boxes of nine.

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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.

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

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Use the clues about the shaded areas to help solve this sudoku

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Each of the main diagonals of this sudoku must contain the numbers 1 to 9 and each rectangle width the numbers 1 to 4.

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

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The puzzle can be solved with the help of small clue-numbers which are either placed on the border lines between selected pairs of neighbouring squares of the grid or placed after slash marks on. . . .

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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.