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Investigate the number of hidden faces when cubes are joined in different ways.

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Introduction

GCSE Mathematics                Hidden Faces Coursework

Hidden Faces Investigation

Investigate the number of hidden faces when cubes are joined in different ways

I am going to investigate the number of hidden faces when cubes are joined in different ways. The aim of this task is to find a formula which is common with every cube and hidden face.

For this investigation I will need to find the formula, which can determine the outcome for the number of hidden faces on ‘n’ cubes. I will start this investigation by drawing cubes that join together in rows also those that are joined to form a cuboid. I will draw up a table to show my results, from which I will hopefully be able to find a pattern that will allow me to express a formula that can be used to find the number of hidden faces when cubes are joined in any form i.e. rows and cuboids.

Task 1

When placing a single cube on a flat surface only 1 face is hidden out of 6. Fundamentally only 5 faces are visible.

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image06.pngimage01.png

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Cubes are a three-dimensional shape with six equally-sized square surfaces; if you multiply the number of cubes by 6 (as there are six faces on a cube) the outcome you will get is the total number of faces.

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Middle

Formula Three in theory is not much efficient and can be time consuming. It will be very difficult and time consuming for example for a person who needs to find the total number of hidden faces for 35 cubes in a row. They would need to draw or build the cubes, count the number of visible faces and then use formula three to find the number of hidden faces.

Therefore I am going to try and find a pattern between the ‘number of cubes’ and the ‘number of hidden faces’, by using the data in table 1.1.

Number of cubes

Number of hidden faces

1                 0   image04.pngimage02.png

2                 +2image03.pngimage14.pngimage04.pngimage02.png

3                 +4image02.pngimage05.png

4                 +6image02.pngimage04.png

5                 +8image04.pngimage02.png

6                 +10image02.png

1image07.png

4image09.pngimage08.png

7image07.png

10image09.png

13

16

Let n be the ‘n’ be the number of cubes:

Experimenting to find a formula for hidden faces for (a) cube(s).

Try 2n – 2

(2 * 1) – 2 = 0 formula does not work

Try 2n + 2

(2 * 1) + 2 = 4 formula does not work

Try 3n + 2

(3 * 1)  + 2 = 5 formula does not work, however the answer given is the same value

for the visible faces (referring to table 1.1)

Try for more number of cubes

Let n = 2

(3*2) + 2 = 8                 correct

Let n = 3

(3*3) + 2 = 11                correct

Let n = 7

(3*7) + 2 = 21                correct

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Conclusion

Formula C:                6xyz – (2yz + 2xz + xy)

Using figure 1 (30 cubes: cuboid) I am going to put this formula to test.

x = 3,  y = 5,  z = 2

Total number of hidden faces =         (6x3x5x2) - (2x5x2 + 2x3x2 + 3x5)

                                =        180 – ( 20 + 12 + 15)

                                =         180 – 47

                                =        133

Number of Cubes in a Cuboid

Number of Visible (seen) faces

Number of Hidden faces

Total Number of faces

12

18

24

30

36

26

33

40

47

54

46

75

104

133

162

72

108

144

180

216

<Data obtained by using diagrams in appendix two>

The following are sequences that I found in the above table:

Number of cubes in a cuboid:                12, 18, 24, 30, 36….. value  + 6 each time.

Number of visible faces:                26, 33, 40, 47, 54….. value + 7 each time.

Number of hidden faces:                46, 75, 104, 133, 162…. value + 29 each time

Sequence found between number of cubes in a cuboid and the number of visible faces:

12 – 26        +14

18 – 33        +15

24 – 40        +16

30 – 47        +17

36 – 54        +18

These are the patterns that I have obtained from Table 1.2, however in regard to my formula for the total number of faces, visible faces and hidden faces I do not see any use of them being used.

Task 2 was accomplished as well successful by using formulas/rules that were applied in Task 1. I basically used formulas from Task 1 as a base to produce formulas in Task 2.

Counting either in a drawing or on a model for faces on a cube(s)/cuboid can be very time consuming. My report gives all the necessary formulas that can be used by anyone for finding out the total number of faces, visible and hidden on any cube(s)/cuboids.

Ali Hayat: 2076                Ealing And West London College: 12420

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