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

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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.

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

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

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

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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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Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

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Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

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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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What is the best way to shunt these carriages so that each train can continue its journey?

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How will you go about finding all the jigsaw pieces that have one peg and one hole?

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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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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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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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This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

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Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

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In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

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Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

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How many different triangles can you make on a circular pegboard that has nine pegs?

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Place six toy ladybirds into the box so that there are two ladybirds in every column and every row.

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Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

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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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Design an arrangement of display boards in the school hall which fits the requirements of different people.

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Jack has nine tiles. He put them together to make a square so that two tiles of the same colour were not beside each other. Can you find another way to do it?

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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 work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?

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First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.

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

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Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?

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

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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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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 practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

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Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

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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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How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?