Linear Equation
Y = mx + c
Nth = mn + c
Nth = 3n
= 3n- 2
I will now use the linear rule on the results above (hidden Faces), I will see if I could find the global formula that will work on any number of cubes in a row.
The sequence goes up in 3’s,
So m = 3
Nth = 3n
So c = -2 FORMULA CHECK 3n - 2
3 × 10 – 2 =28
This shows that the formula is correct. I now know that the formula works because I have tried it with a number out side of the sequence.
Example
1 + 2 (n - 1)
2 + 2 (2 - 1)
2 + 2 = 4
This is to show how many gaps are created when the cubes are stocked horizontally in a row.
The ‘n’ is equal to the number of cubes. For each cube there is at least one hidden face. When the cubes are stocked side by side they meet at gaps which create more hidden faces. The number of gaps is the numbers of cubes are taken away by. The hidden faces are in between gaps.
I will extend my investigation to consider more rows of cubes put together.
I will want to rack this formula, H/F = T/F – S/F. From this I can derive this formula -
H/F = 6 (L × W × H) – [2 (W× H) +2 (L × H) + W × L]
Total Number of Faces
- There are 6 faces in a cube.
- Get the length, width and height and times it all and the multiply the 6.
- This will then give you the total number of faces.
- Total number of faces is basically 6 × the volume of the cube or cuboid.
Seen Faces
- Multiply the height and width twice
- Multiply the length and height twice
- Multiply width and height
- Add all this together and to will get the seen faces
- Seen faces are basically the surface area of the cube or cuboid.
Hidden Faces
- Get the total number of faces and seen faces and subtract them
- You will then have the answer of hidden faces
L
I will now explain it more by using this example.
For this you will do-
H/F = T/F – S/F
H/F = 6 (L × W × H) – [2 (W× H) +2 (L × H) + W × L]
H/F = 6 × 6 × 2 × 2 - 8 + 24 + 12
T/F = 144 CM3
S/F = 44 CM
H/F = 100 CM3
Conclusion
From this we have found out the global formula and to use it and explain it properly. I have also found out another formula which helps us to find out the hidden faces for a cube or a cuboid.
Global formula
From the information which I have put on the first few pages we can see it explains the global formula and when cubes are stacked horizontally you get the formula of 3n-2. Even it is not part of a sequence
I have also found out a formula for finding out how to find the number of hidden faces in a cube or a cuboid just by subtracting total number of faces with the number of seen faces.
I used the following formula -
H/F = T/F – S/F
H/F = 6 (L × W × H) – [2 (W× H) +2 (L × H) + W × L]
By Nahidur Rahman 10o Ma7
Mathematics- Hidden Faces coursework
Mr Bailey