Hidden Faces.

HIDDEN FACES

Hidden Faces: Hidden faces are those hidden after the cubes have been viewed from all angles.

Introduction: I am investigating the number of hidden faces for other cuboids made from cubes. I will use visual representation to display my results in the form of graphs. I will collect my results in a table. I will start to collect my information in my table starting with one cube and building them up into rows and different sized cuboids. At the end of my investigation I hope to have a formula worked out, and also I hope to be able to find the number of hidden faces on a cuboids made up from 30 cubes.

Collecting Data: I have drawn a table to record my results. In the first column I have the number of cubes and in the second I have the number of hidden faces. In my table I have found the hidden faces for every one cube put down there is one hidden face on the bottom. And if a cube is put is put next to another there are always two hidden faces between them.

What This Shows: My results show a pattern occurring. For every cube that is added the number of hidden faces grows by three each time. I my table the number of hidden faces are lade out in a sequence. This will help me to find a formula so I can work out the number of hidden faces without having to count them each time. To show how I know the numbers are in sequence I will use my knowledge of methods of difference.

1st 2nd 3rd 4th 5th 6th 7th

1 4 7 10 13 16 19

+3 +3 +3 +3 +3 +3

The Differences: In this sequence (using the number of hidden faces from my table) the difference between them is +3. So what ever my formula turns out to be it must have a 3 in it. If I use 'N' as representing the number of cubes. My formula looks like this '3N' or 3xN. So If a number of cubes are put into my formula it will go through like this.

X =

Prediction: I predict that if I put a 9 and then a ...

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1st 2nd 3rd 4th 5th 6th 7th

1 4 7 10 13 16 19

+3 +3 +3 +3 +3 +3

X =

Prediction: I predict that if I put a 9 and then a 7 into my formula. 9 cubes will have 27 hidden faces. And if I put 7 cubes into my formula there will be 21 hidden faces.

The Final Piece: If I put a 9 into my formula it gives me 3x9=27. And if I put a 7 in it will come out like this 3x7 =21. So my prediction was right but if you count the number of hidden faces without using the formula 9 cubes has 25 hidden faces and 7 cubes has 19 hidden faces. So each time the number of hidden faces is -2 more than I predicted. This is the final piece of the formula. Now my formula looks like this 3n-2

Proof: To prove that my formula works I am going to put a larger number of cubes to support my theory

10 cubes = 3 x 10 = 30

30 - 2 = 28 hidden faces

15 cubes = 3 x 15 = 45

45 - 2 = 43 hidden faces

20 cubes = 3 x 20 = 60

60 - 2 = 58 hidden faces

Another Way: Another way of working out the number of hidden faces. If you take the number of side on a cube which is 6, and then multiply them by the number of cubes, then subtract them from the number of side you can see you get the number of hidden faces. This can also be put into a formula it looks like this 6n - s.

So my formula is correct and accurate and works for any number of cubes in a row. Now I going to find out the number of hidden faces for cubes stacked up on each other.

2 High: Now I am going to see if my formula works for 2 cubes stacked on top of each other and another two stacked next to that.

Does My Formula Work: My formula for the number of cubes in a row is 3n-2. To see if this works I am going to input some of the number of cubes form my new table above, to see if the old formula works.

3 sets of cubes = 3 x 6 = 18

18 - 2 = 16

5 sets of cubes = 3 x 10 = 30

30 - 2 = 28

7 sets of cubes = 3 x 14 = 42

42 - 2 = 40

So my formula dose not work for two cubes stacked one on top of the other. I know need to find another formula to help me work out the number of hidden faces. I will do this using my knowledge of methods of differences. Shown in this sequence.

2nd 4th 6th 8th 10th 12th 14th

3 10 17 24 31 38 45

+7 +7 +7 +7 +7 +7

Second Formula: In my second sequence my the number of cubes seems to go up in sevens each time. But because my cubes are going up in two's instead of ones this time, I need to find out what half of seven is to give me the first bit of my formula. So my first piece of my formula is '3.5'. Then I am going to use 'N' as representing the number of cubes my formula looks like this '3.5n'.

Prediction: I predict that the number of hidden faces for 16 cubes will be 52 hidden faces. And the number of hidden faces for 18 cubes will be 59 hidden faces. To see if I am right I am going to use my formula so far to test these predictions.

16 cubes stacked = 3.5 x 16 = 56

18 cubes stacked = 3.5 x 18 = 63

So my formula so far is wrong. Because each time I put in a number of cubes the number of hidden faces is four more than I predicted. Each time I am having to - 4 each time to get the right number of hidden faces. My formula is now complete and looks like this 3.5n - 4.To prove this I am going to input different amounts of cubes.

26 stacked cubes = 3.5 x 26 = 91

91 - 4 = 86 hidden faces

30 stacked cubes = 3.5 x 30 = 105

105 - 4 = 101 hidden faces

36 stacked cubes = 3.5 x 36 = 126

126 - 4 = 122 hidden faces

So my second formula also works. My second way of working it out like I did for cubes in a row which is 6n - S this also works using stacked cubes

Conclusion: In conclusion my investigation shows how my formulas help to find the number of hidden faces. My first formula works only for cubes in a row. My second formula only works for cubes stacked two high in a line. But the amazing thing is that the formula that works for both is the other way of working out the number of hidden faces 6n- S. This formula involves a lot more working out but can be effective if used.

Hidden Faces.

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