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

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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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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?

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

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

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

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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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A few extra challenges set by some young NRICH members.

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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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Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

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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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Have a go at this game which has been inspired by the Big Internet Math-Off 2019. Can you gain more columns of lily pads than your opponent?

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

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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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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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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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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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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.

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

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

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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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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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Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?

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

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

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Add or subtract the two numbers on the spinners and try to complete a row of three. Are there some numbers that are good to aim for?

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A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.

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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.