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I will be investigating different patterns that can be found in cubes that are constructed from smaller 1cm cubes and are painted on the outside.

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Introduction

CUBES I will be investigating different patterns that can be found in cubes that are constructed from smaller 1cm� cubes and are painted on the outside, when these smaller cubes are taken apart some are only partly painted others none at all. Here is a table showing the number of faces painted. Cube Number Cube length No. Of small cubes No. Of small cubes with 3 painted faces No. Of small cubes with 2 painted faces No. Of small cubes with 1 painted faces No. Of small cubes with 0 painted faces 2 2*2*2 8 8 0 0 0 3 3*3*3 27 8 12 6 1 4 4*4*4 64 8 24 24 8 Straight away I noticed that the number of cubes with three sides painted is always 8 as seen in the table below and this is true for all cubes except one (see exceptions). ...read more.

Middle

then noticed that it would make sense to minus two from the cube number as for the first two cubes there aren't any cubes with two sides painted. As seen below in blue I get a matching pattern that forms the twelve times table. Simply by knowing the dimensions of a cube I can work out how many small cubes there would be with two sides painted in any cube e.g. the following: 3 -2= 1*12=12 4 -2= 2*12=24 5 -2= 3*12=36 6 -2= 4*12=48 7 -2= 5*12=60 8 -2= 6*12=72 Thus finding and proving the formula Y=(X-2)12 This table shows the number of cubes with only one side painted. Cube Number (X) Cube dimensions No. Of small cubes No. Of small cubes with 1 painted faces 2 2*2*2 8 0 3 3*3*3 27 6 4 4*4*4 64 24 5 5*5*5 125 54 6 ...read more.

Conclusion

Cube Number (X) Cube dimensions No. Of small cubes No. Of small cubes with 0 painted faces 2 2*2*2 8 0 3 3*3*3 27 1 4 4*4*4 64 8 5 5*5*5 125 27 6 6*6*6 216 64 This formula was perhaps the easiest to puzzle out, because of all the clues I have learnt to spot in previous formulas. The first thing I noticed about this is that the number of small cubes with no sides painted mirrored that of the number of total small cubes starting at eight two rows down in red, this led me to believe that a minus of two was needed in the formula. I then noticed that the number of small cubes were obviously all made up of cubed numbers so the formula should also involve this, this is what I came up with: Y= 3(x-2) 3(2-2)=0 3(3-2)=1 3(4-2)=8 3(5-2)=27 3(6-2)=64 This was the proof of this formula, corresponding to the above tables results. ...read more.

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