Now I need to find out the value of c so I can substitute it into the equation. To find c I will chose a number of cubes from the table and its results and place it into the equation.
e.g. tn=an+c
13=3(5)+c
13=15+c
Now all I have to do is rearrange the formula so I can find out c.
13=3(5)+c
13-15=c
c=-2
tn=3n+(-2)
the equation is not in its simplest form so now I need to multiply out the brackets so I can get my final formula.
tn=3n-2
To see whether or not the formula works I will test it using a number of cubes which Is not in my table above. I will use 10 cubes for this.
tn=an+c
tn=3n-2
tn=3(10)-2
tn=30-2
tn=28
As I can see from the above equation I found out that my formula for hidden faces in one row has worked.
For the next part of my coursework I need to generate another formula but for a different structure of cubes. I will be using 2 rows for this part of the coursework.
2 rows
2 cubes
12 faces
4 hidden Faces
2 rows
4 cubes
24 faces
12 hidden faces
2 rows
6 cubes
36 faces
20 hidden faces
Cubes In A Row Total Faces Seen Faces Hidden Faces
2x1 12 8 4
2x2 24 12 12
2x3 36 16 20
2x4 48 20 28
2x5 60 24 36
2x6 72 28 44
2x7 84 32 52
2x8 96 36 60
12n 4n+4 8n-4
The graph above shows us the number of cubes, faces a hidden faces. It also shows the formulae for finding out the number of hidden faces, total faces and number of faces seen.
Above shows the relationship between the number of cubes and the number of faces.
By looking at the graph and the chart I have generated my second equation to find the number of hidden faces in 2 rows.
The table below shows the relationship between the number of cubes and the number of hidden faces. I did not draw the other two tables because they were not relevant in finding out the global formula.
nth term 2 4 6 8 10 12 14 16
tn 4 12 20 28 36 44 52 60
1st diff + 8 + 8 + 8 + 8 + 8 + 8 + 8
Formula for working out the total number of hidden faces:
This pattern also has only the 1st difference so I could see that this was going to be a linear equation as well.
So again I would have to follow the rule:
tn = an + c
As you know tn is total number of hidden faces and n is the number of cubes, so the two unknowns are once again a and c.
a has been replaced with 8 as that represents the 1st difference, therefore the equation now looks like: tn = 8n+c.
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.
To make sure my formula works I will have to test it out first. I will use a number which is not part of my table. I will use 30 cubes.
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.
. 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
The table above shows the total number of faces on an ‘n’ number of cubes. As we increase the number of cubes being added on the number of faces increases by 6. The formula to find out the total number of faces is: 6n
E.g. 4 is the nth term so you multiply 4 by 6, which gives you a total of 24 which is the answer.
Nth term 1 2 3 4 5 6 7 8
Seen faces 5 8 11 14 17 20 23 26
difference + 3 + 3 + 3 + 3 + 3 + 3 + 3
The table above shows the amount of faces seen on an ‘n’ number of cubes. The formula for working out the number of faces which can be seen is: 3n+2
E.g. 3 is the nth term so you have to multiply 3 by 3 3(3)+2
Which gives a total of 9. you then add 2 which gives a final total of 11.
Below shows the relationship between the cubes and the number of faces. Both hidden and seen.
Nth term 1 2 3 4 5 6 7 8
Hidden faces 1 4 7 10 13 16 19 22
differences + 3 + 3 + 3 + 3 + 3 + 3 + 3
The graph above shows how many hidden faces there are related to the number of cubes.
The graph and the table above shows the relationship between the number of cubes and the number of faces seen and unseen. Both the graph and the table above will now allow me to work out the formula for the number of hidden faces in one row.
To find the global formula for the number of hidden faces in one row I have to refer to the table above. As you can see from the table it will be a linear equation because there is only 1 line of difference. The general linear equation is
...
y=mx+c
Therefore the linear rule is in the form of
tn=an+c
In the equation tn is the total number of hidden faces and n is the number of cubes. Therefore I need to find out the equation for a and c are. In the equation a is equal to the first difference. So I can replace the a with a 3, which makes
tn=3n+c
Now I need to find out the value of c so I can substitute it into the equation. To find c I will chose a number of cubes from the table and its results and place it into the equation.
e.g. tn=an+c
13=3(5)+c
13=15+c
Now all I have to do is rearrange the formula so I can find out c.
13=3(5)+c
13-15=c
c=-2
tn=3n+(-2)
the equation is not in its simplest form so now I need to multiply out the brackets so I can get my final formula.
tn=3n-2
To see whether or not the formula works I will test it using a number of cubes which Is not in my table above. I will use 10 cubes for this.
tn=an+c
tn=3n-2
tn=3(10)-2
tn=30-2
tn=28
As I can see from the above equation I found out that my formula for hidden faces in one row has worked.
For the next part of my coursework I need to generate another formula but for a different structure of cubes. I will be using 2 rows for this part of the coursework.
2 rows
2 cubes
12 faces
4 hidden Faces
2 rows
4 cubes
24 faces
12 hidden faces
2 rows
6 cubes
36 faces
20 hidden faces
Cubes In A Row Total Faces Seen Faces Hidden Faces
2x1 12 8 4
2x2 24 12 12
2x3 36 16 20
2x4 48 20 28
2x5 60 24 36
2x6 72 28 44
2x7 84 32 52
2x8 96 36 60
12n 4n+4 8n-4
The graph above shows us the number of cubes, faces a hidden faces. It also shows the formulae for finding out the number of hidden faces, total faces and number of faces seen.
Above shows the relationship between the number of cubes and the number of faces.
By looking at the graph and the chart I have generated my second equation to find the number of hidden faces in 2 rows.
The table below shows the relationship between the number of cubes and the number of hidden faces. I did not draw the other two tables because they were not relevant in finding out the global formula.
nth term 2 4 6 8 10 12 14 16
tn 4 12 20 28 36 44 52 60
1st diff + 8 + 8 + 8 + 8 + 8 + 8 + 8
Formula for working out the total number of hidden faces:
This pattern also has only the 1st difference so I could see that this was going to be a linear equation as well.
So again I would have to follow the rule:
tn = an + c
As you know tn is total number of hidden faces and n is the number of cubes, so the two unknowns are once again a and c.
a has been replaced with 8 as that represents the 1st difference, therefore the equation now looks like: tn = 8n+c.
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.
To make sure my formula works I will have to test it out first. I will use a number which is not part of my table. I will use 30 cubes.
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.
