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By Becky and Lydia 80 The IN.BB.LG team

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Names: Becky and Lydia. Age: Both 12 years old. School: Settlebeck High School, Sedburgh, Cumbria, England Project: Painted Cube Problem

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Isaac Newton was an English mathematician, who discovered the binomial theorem, (A theory he came up with) he also invented calculus, and produced theories of mechanics, optics, and the law of universal gravitation. Many of his ideas for which he is famous were developed in isolation in the year 1665 during the Great Plague. He was also knighted Sir Isaac Newton.

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For this slideshow you will need: Isometric paper, Pencil, Four different coloured pencil crayons, Calculator, AND YOUR BRAIN!

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For our first problem we are going to show you is the 1 by 1 by 1 cube. So to build the 1 by 1 by 1cube all you need is 1 cube. Then all you need to do is to count the number of cubes which have all of it’s faces panted. This cube has 6 faces painted. All the cubes on this cube, all have 3 faces painted. For our Second we are going to show you is the 2 by 2 by 2 cube. So to build the 2 by 2 by 2 cube all you need to do is get 8 cubes and put them together like the diagram above. Then all you need to do is count how many cubes have three faces painted.

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Cubes with 0 faces Painted Cubes with 1 Face Painted Cube s with 2 Faces Paint ed Cubes with 3 Faces Painted Cubes with 6 Faces Painted Total number of cubes 1 by 1 by 1 000011 2 by 2 by 2 000808 3 by 3 by 3 Answers will be shown Later on in the slideshow 4 by 4 by 4

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Counting the faces on each individual cube sounds easy but it’s not. If you wont it to be easy follow these simple steps. Draw your cube on isometric paper. Colour all the cubes which all have three faces painted blue. Colour all the cubes which all have two faces painted green. Colour all the cubes which all have one face painted brown. Remember if on the inside of the cube there are cubes with no faces painted to count them ones. Remember you can not see one of the cubes which has three faces painted because you can not draw it on a piece of isometric paper. Colouring in represents painting the the cube.

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3 faces painted 2 Faces painted 1 face painted 0 Faces painted

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Here is a 3x3x3 cube. In a 3 x 3 x 3 cube there are: 1 cubes with 0 faces painted. 6 cubes with 1 face painted. 12 cubes with 2 faces painted. 8 cubes with 3 faces painted. In a 4 x 4 x 4 cube there are: 8 cubes with 0 faces painted. 24 cubes with 1 face painted. 24 cubes with 2 faces painted. 8 cubes with 3 faces painted. Here is a 4x4x4 cube

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Cubes with 0 faces Painted Cubes with 1 Face Painted Cube s with 2 Faces Paint ed Cubes with 3 Faces Painted Cubes with 6 Faces Painted Total number of cubes 1 by 1 by 1 000011 2 by 2 by 2 000808 3 by 3 by 3 16128027 4 by 4 by 4 824 8064

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An exploded cube is a really simple way of working out the painted cube problem. First you explode a cube like the diagram at the bottom. Then you colour all the sides if they are 3 faces painted sides or 2 faces painted sides or if they are 1 face painted side or if they are 0 faces painted sides. This will help you by making the painted cube problem a lot easier. You try and draw an exploded cube on your isometric paper. This is the centre cube which does not get painted but we have just highlighted it to make it easier to see

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This cube has three of its faces painted. These are all cubes which have two of there faces painted. All these cubes have one of there faces painted.

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Work out how many cubes have : 3 faces painted 2 faces painted 1 face painted 0 faces painted Work out how many cubes have : 3 faces painted 2 faces painted 1 face painted 0 faces panted Here is a 6 by 6 by 6 cube Here is a 5 by 5 by 5 cube

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The answer to the 5 by 5 by 5 panted cube is 8 cubes have 3 faces painted 36 cubes have 2 faces painted 54 cubes have 1 face painted 27 cube has 0 faces painted The answer to the 6 by 6 by 6 8 cubes have 3 faces painted. 48 cubes have 2 faces painted. 96 cubes have 1 face painted. 64 cubes have 0 faces painted. Were you right!

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How many cubes have: 3 faces painted 2 faces painted 1 face painted 0 faces painted Total number of cubes 27 This is an 3 by 3 by 3 exploded cube.

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This is a 4 by 4 by 4 exploded cube. How many cubes have: 3 faces painted 2 faces painted 1 face painted 0 faces painted Total number of cubes

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The answers to the last two exploded cube problems. In a 3 x 3 x 3 cube there are: 1 cubes with 0 faces painted. 6 cubes with 1 face painted. 12 cubes with 2 faces painted. 8 cubes with 3 faces painted. The answers to the 3x3x3 cube are: The answers to the 4x4x4 cube are: In a 4 x 4 x 4 cube there are: 8 cubes with 0 faces painted. 24 cubes with 1 face painted. 24 cubes with 2 faces painted. 8 cubes with 3 faces painted. Hope you got it right!!!!!

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To work out the total number of cubes in a cube lets say it was a 10 by 10 by 10 cube. You would do 10 x 10 x 10 that answer would be how many cubes are in a10 by10 by 10 cube. To work out the total number of cubes that have 3 faces painted in any cube is very easy because all cubes have 8 cubes with 3 faces painted, if the cube dose not have 8 cubes with 3 faces painted it is not a cube. To work out the total number of cubes have 2 faces painted in any cube, you find out how many cubes are along the edge of one side then takeaway 2 then times that number by 12 because there are 12 edges on a cube. To work out the total number of cubes with 0 faces painted, you takeaway 2 from the 10. Then you do 8*8*8. To work out the total number of cubes which have 1 face painted you, work out how many cubes are on a face by doing 10-2 then you square that number. Then you times that number by 6 because there are 6 faces on a cube.

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We both hope you have enjoyed our side show and hope it has taught you some things about the painted cube problem! Goodbye and thank you for joining us! ^Click to go back^ to the start

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