MATHS COURSEWORK
HIDDEN FACES.
Aim:
The aim of this coursework is to find a global formula for the total number of hidden faces for any number of cubes in rows.
A cube has six faces in total. Hidden faces are faces that cannot be seen when a cube is placed on a table or in rows along side other cubes. If you place five cubes along side each other on to a table, they have a total of 30 faces of which 13 faces are hidden and 17 can be seen.
In order to find the global formula I will have to find general formulae for the different number of rows by producing tables and drawing diagrams.
I will first find out a general formula for one row of cubes. I will start at one cube and go up to eight cubes in a row.
Results:
Cubes in a row
Total faces
Faces seen
Faces hidden
x1
6
5
x2
2
8
4
x3
8
1
7
x4
24
4
0
x5
30
7
3
x6
36
20
6
x7
42
23
9
x8
48
26
22
6n
3n+2
3n-2
In the table and graph above I have shown the relationship between the cubes, the total number of faces, their hidden faces and the faces that can be seen. For me to find out the general formula I will have to do one more table for the number of hidden faces.
Number Of Cubes.
2
3
4
5
6
7
8
Hidden Faces.
4
7
0
3
6
9
22
st Difference
+
3
+
3
+
3
+
3
In the table above there is only one line of difference, which tells me that it is a linear equation
The general form of a linear equation is:
y=mx+c
Therefore the linear rule is in the form of:
tn=an+c
tn=3n+c
In the above equation, the total number of hidden faces is tn and n is the number of cubes. I have replaced a to 3 because a is equal to the first difference in the above table. There fore the only unknown value is the letter c.
HIDDEN FACES.
Aim:
The aim of this coursework is to find a global formula for the total number of hidden faces for any number of cubes in rows.
A cube has six faces in total. Hidden faces are faces that cannot be seen when a cube is placed on a table or in rows along side other cubes. If you place five cubes along side each other on to a table, they have a total of 30 faces of which 13 faces are hidden and 17 can be seen.
In order to find the global formula I will have to find general formulae for the different number of rows by producing tables and drawing diagrams.
I will first find out a general formula for one row of cubes. I will start at one cube and go up to eight cubes in a row.
Results:
Cubes in a row
Total faces
Faces seen
Faces hidden
x1
6
5
x2
2
8
4
x3
8
1
7
x4
24
4
0
x5
30
7
3
x6
36
20
6
x7
42
23
9
x8
48
26
22
6n
3n+2
3n-2
In the table and graph above I have shown the relationship between the cubes, the total number of faces, their hidden faces and the faces that can be seen. For me to find out the general formula I will have to do one more table for the number of hidden faces.
Number Of Cubes.
2
3
4
5
6
7
8
Hidden Faces.
4
7
0
3
6
9
22
st Difference
+
3
+
3
+
3
+
3
In the table above there is only one line of difference, which tells me that it is a linear equation
The general form of a linear equation is:
y=mx+c
Therefore the linear rule is in the form of:
tn=an+c
tn=3n+c
In the above equation, the total number of hidden faces is tn and n is the number of cubes. I have replaced a to 3 because a is equal to the first difference in the above table. There fore the only unknown value is the letter c.
