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Hidden faces

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Introduction

Hidden faces Introduction I am investigating the number of hidden faces for different rows of cubes. When the cubes are placed on a table, you cannot touch them or move them however you may move the table around. Examples When a cube is placed on a table only 5 of the faces can be seen. So 1 face is hidden. Figure 1 on the dotted paper shows this. When five cubes are placed on a table only 17 faces can be seen. So 13 faces are hidden. Figure 2 on the dotted paper shows this. Part 1 I will tackle this problem by investigating different numbers of cubes. These cubes that I'll be using are 1 cube, 2 cubes, 3 cubes, 4 cubes and 5 cubes. These cubes are shown on the dotted paper. This method is sensible because using the number of cubes I'd be using could lead to finding a formula easily. Number of cubes Number of faces seen Number of faces hidden Total number of faces 1 5 1 6 2 8 4 12 3 11 7 18 4 14 10 24 5 17 13 30 Patterns Patterns that I noticed was in the column 'Number of faces seen' that when a cube has 5 faces seen, 2 cubes has 8 faces seen. ...read more.

Middle

So I thought that could help in the formula. Next I looked at the column 'Number of cubes and I thought this could lead to success, which it nearly did. When I multiplied 3 by the number of cubes in the row, it came out as the formula was 2 to many so I then subtracted 2 to get the correct formula as I checked it with the other numbers of cubes in the row. I counted the hidden faces by counting the number of cubes because that's the amount hidden on the bottom. I then counted the hidden faces in between the cubes. Between the cubes, which were joined together with another cube was 2 hidden faces. So I multiplied 2 by the number of cubes, which were joined together with another cube. With this formula I could explore different number of cubes in a row. Number of cubes Number of faces seen Number of faces hidden Total number of faces 10 32 28 60 20 62 58 120 30 92 88 180 40 122 118 240 50 152 148 300 Part Two I am now going to investigate the number of hidden faces for different cuboids made from cubes. ...read more.

Conclusion

A cuboid of 42 cubes has a total of 246 faces, a cuboid of 48 cubes has a total of 279 faces and a cuboid of 54 cubes has a total of 312 faces. Between each cuboid is a gap of 33. This could help me with a finding of a formula. Formula The formula that I have found for the number of hidden faces for different cuboids made from cubes is I will count the cubes by counting the number of faces seen on the top of the cuboid to give me the number of faces hidden on the bottom. I will then multiply that by 3 because there is another layer of cubes, where the cubes are joined together are 2 faces. So you'll have to add them to the bottom of the number of faces hidden to the number of faces hidden in the second layer. I then looked at the top layer of the cuboid to see how many faces are hidden. I counted the cubes going horizontal and multiplied it by 2 and subtracted 2 because the faces on the end were seen not hidden. I multiplied this by 3 because there are 3 layers that are vertical. I then counted the vertical cubes and subtracted by 2 because the faces on the end were seen not hidden. I multiplied this by 5 because there are 5 layers horizontal. ...read more.

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