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Painted Cube Ivestigation.

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Introduction

Painted Cube Ivestigation This investigation was about investigating what happened when a large cube made up of smaller cubes was put into a pot of paint. First, I labelled the smaller cubes according to their position in the large cube. = Corner Cube = Middle Cube = Core Cube = Central Cube This is my diagram of a 3 x 3 x 3 cube. The Cubes are labelled. If the large cube were covered in paint, not every face on every small cube would be covered. Some cubes, according to position, would have only 1 face painted, some would have 2 faces painted etc. I first investigated a 33 cube. These are my results. Total in Cube (33) Faces Covered (each) Total faces Covered Corner Cubes 8 3 24 Core Cubes 1 0 0 Middle Cubes 12 2 24 Central Cubes 6 1 6 Total Faces in Cube Complete Cube 27 162 54 They show: * The total number of cubes of each type in a large cube (33). ...read more.

Middle

Total in Cube (53) Faces Covered (each) Total faces Covered Corner Cubes 8 3 24 Core Cubes 27 0 0 Middle Cubes 36 2 72 Central Cubes 54 1 54 Total faces in cube Complete Cube 125 750 150 My corner cube prediction is correct. When n ? 2, the number of corner cubes is always 8. From my result tables, I can also deduce the core cube number = (n-2) 3. Total in Cube (63) Faces Covered (each) Total faces Covered Corner Cubes 8 3 24 Core Cubes 64 0 0 Middle Cubes 48 2 96 Central Cubes 96 1 96 Total faces in cube Complete Cube 216 1296 216 This graph shows the faces painted against their value of n (Red Line). The line is a curve becoming flatter which shows that the number of faces painted is associated with a power of n. This graph also shows n against n3 (Blue line). This tells me that the power of n associated with the number of faces painted is n2 because the first graph's curve was less curved than this and the higher the power the more curved the graph line will become. ...read more.

Conclusion

6((n-2)2 This formula is the same for working out the number of faces painted because each central cube has 1 face painted. We will call the total number of middle cubes m. n=3 n=4 n=5 n=6 m=12 m=24 m=36 m=48 n=3 n=2 m=12 m=0 From the second line of data I can tell that m = 12(n-2). This is because when n=2, m=0. This tells me that the equation must be (n-2) multiplied by something, because for every n value above 2, m is above 0. When n=3, m=12 so the equation must be 12(n-2) because 3-2 = 1 and 1x12 = 12. To work out the number of faces painted on all the middle cubes is this formula multiplied by 2 because each middle cube has 2 exposed faces. 2(12(n-2)) To Present all my formulae I will make a table with values for n3. Total in Cube (n3) Faces Covered (each) Total faces Covered Corner Cubes 8 3 24 Core Cubes (n-2)3 0 0 Middle Cubes 12(n-2) 2 2(12(n-2)) Central Cubes 6((n-2)2 1 6((n-2)2 Total faces in cube Complete Cube n3 6n3 6n2 Given more time on this investigation I would have further investigated 3-D shapes other than cubes. 1 1 ...read more.

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