# Hidden Faces

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Introduction

Hidden Faces Coursework

A cube a total of 6 sides, when it is places on a surface only 5 of the 6 faces can be seen. However if you place 5 cubes side by side, there is a total of 30 faces, but out of this 30 only 17 can be seen.

In this coursework I will be finding out the Hidden Faces Coursework

A cube a total of 6 sides, when it is places on a surface only 5 of the 6 faces can be seen. However if you place 5 cubes side by side, there is a total of 30 faces, but out of this 30 only 17 can be seen.

In this coursework I will be finding out the global formula for the total number of hidden faces for any number of cubes in any way positioned. To find this out I will be testing various numbers of cubes in different positions. This will enable me to find out several different formulae. Using the formulas found I will then be able to find out the global formula. I am generating only 3 formulae to get to the global formula.

1 row 6 faces

1 cube 1 hidden face

1 row 12 faces

2 cubes 4 hidden faces

1 row 18 faces

3 cubes 7 hidden faces

1 row 24 faces

4 cubes 10 hidden faces

1 row 30 faces

5 cubes 13 hidden faces

1 row 36 faces

6 cubes 16 hidden faces

1 row 42 faces

7 cubes 19 hidden faces

1 row 48 faces

8 cubes 22 hidden faces

From the cubes drawn above I can see a pattern being formed. The number of hidden faces goes up by 3 every time a cube is added on the end.

Cubes in a row Total faces Faces seen Faces unseen

1x1 6 5 1

1x2 12 8 4

1x3 18 11 7

1x4 24 14 10

1x5 30 17 13

1x6 36 20 20

1x7 42 23 19

1x8 48 26 22

The graph above show the number of hidden faces, the number of faces which can be seen and the total number of faces.

Nth term 1 2 3 4 5 6 7 8

Total faces 6 12 18 24 30 36 42 48

difference + 6 + 6 + 6 + 6 + 6 + 6 + 6

Middle

Again two methods can be used to work out the c term. The first way, which can be used, is to use the zero term:...

so as the 1st difference is +3 the working out would be 4 – 8 = -4.

Another method that could be used is, you pick any number from the table above and it’s result, and work it out as an equation:

tn = 8n+c

44 = 8(6)+c

44 = 48+c

You need to rearrange the formula so c is on it’s own, and change the + to -.

c = 44 – 48

c = -4

So the second general formula would look like: tn = 8n - 4

To make sure my formula is right I will test it but I will use a value of cubes which is not in my table. I will use 20 cubes.

Now all I have to do is substitute the information found into the equation.

The final part of my investigation is to generate another formula, all these three formulae should help me find out the global formula.

To get my third and final formula I will investigate using 3 rows of cubes. This should hopefully help me get the global formula.

Cubes In A Row Total Faces Seen Faces Hidden Faces

3x1 18 11 7

3x2 36 16 20

3x3 54 21 33

3x4 72 26 46

3x5 90 31 59

3x6 108 36 72

3x7 126 41 85

3x8 144 46 98

18n 5n+6 13n-6

The table above shows the formulae for total faces, hidden faces and seen faces.

The graph above show the relationship between the faces and the number of cubes.

nth term 3 6 9 12 15 18 21 24

tn 7 20 33 46 59 72 85 98

1st diff + 13 + 13 + 13 + 13 + 13 + 13 + 13

The table above shows the difference between the number of hidden faces and the number of cubes. I have to do all the things listed above in the previous investigations, but using the information gained in this investigation.

I have to substitute the letters with numbers.

Conclusion

tn=an+c

tn=13n-6

tn=13(30)-6

tn=390-6

tn=384

The final formula is 13n-6

The final section of my coursework is to find out the global formula for finding out the total number of hidden faces in any given number of cubes and rows.

I know that the formulae for finding out the different surfaces areas are: length multiplied by width, length multiplied by height and height multiplied by width. I also know the 6LWH will give me the volume of the cube, which is the total number of faces in a group of cubes. This information will help me find out the global formula.

As 6LWH is the total number of faces, this can be used as the first part of my formula.

6LWH

Now I need to subtract the faces which are not inside of the cube, to do this I will use the formulae for finding out the surface area.

Firstly I will have to minus the top are of the cubes as these are all visible, the formula for the surface area is length multiplied by width. This makes our formula

6LWH-LW

Now I have to minus the four surface areas that are on the side of the cuboids. two of these will always be equal, as will the other two sides. To find out the surface area of the two different sized sides, I will have to do length multiplied by width for one and width multiplied by height for the other. But because these surfaces are in pairs I will have to multiply them both by two. This will give me the final formula.

6LWH-2HW+2HL-LW

The formula shown above is used to work out hidden faces in any number of cubes in any formation.

This concludes my coursework and I reached my target, which was to find the global formula by only using three formulae.

This student written piece of work is one of many that can be found in our GCSE Hidden Faces and Cubes section.

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