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

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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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.

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

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

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Replace the letters with numbers to make the addition work out correctly. R E A D + T H I S = P A G E

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

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

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

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

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By selecting digits for an addition grid, what targets can you make?

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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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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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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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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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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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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 values of the nine letters in the sum: FOOT + BALL = GAME

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

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

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

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