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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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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?

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.

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.

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

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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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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?

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

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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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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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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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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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Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

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

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

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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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My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

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

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

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

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If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

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

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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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A few extra challenges set by some young NRICH members.

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

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You have been given nine weights, one of which is slightly heavier than the rest. Can you work out which weight is heavier in just two weighings of the balance?

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Solve this Sudoku puzzle whose clues are in the form of sums of the numbers which should appear in diagonal opposite cells.

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

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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?